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kitwallace · 4 years

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Herschel Enneahedron, Polyhedral Graphs and George Hart

Reading Matt Parker’s Humble Pi (a very welcome Xmas Gift), I came across a reference to a strange polyhedron developed and described by Christian Lawson-Perfect in his 2013 blog.

The starting point is a polyhedral graph named after the astronomer Alexander Stewart Herschel, grandson of William Herschel of Bath.

Polyhedal graphs

I was only vaguely aware of polyhedral graphs. Given a 3D polyhedron, this is 2D diagram of nodes and edges which abstracts the connectivity of vertices. The graph is planar (i.e no crossings) and three-connected (which I think means that every node must be in at least 3 edges and rules out two edges between the same vertices). The graph can be constructed by projecting the polyhedron onto one of its faces. This face becomes the outer edge of the graph, the remaining faces interior to that (with opposite orientation).

The graph form of the polyhedron makes it easier to explore the properties of paths between vertices. For example we might be interested to know if there is a path which includes every node once and once only (called a Hamiltonian Path) and whether such a path can be closed (i.e. starts and finishes at the same node) - a Hamiltonian Cycle.

Herschel’s graph

Hershel’s graph is the smallest graph for which no Hamilton cycle exists.

(from Christian’s blog)

The question is, what polyhedron does it correspond to? Christian used a geometrical argument to construct the coordinates of the vertices given in his blog. However when combined with the faces as constructed from the graph, these didn’t work. I noted in the comments on Christian’s blog that Bill Owen had constructed the polyhedron in OpenSCAD. However the faces here, as reconstructed from the triangles, did not correspond to those in the graph. It seemed that Bill had switched the pair of nodes (6,5) with (8,9). This is the same as switching the values of nodes 1 and 3, and it is this corrected form which is now on my Polyhedral index, with the ‘height’ parameter chosen so that the three large faces are square.

The polyhedron is shown here in its open-face form:

Graph <> polyhedron mapping

I wondered if a more general approach to mapping between polyhedral graphs and 3D polyhedron was possible. I found a mention of an approach developed by George Hart but it was lost in the mists of Mathematica.However knowing George’s work I guessed that it involved canonicalisation as used to construct the dual of a polyhedron.

If I could create a bi-directional mapping between graph and 3D polyhedron, it would not only save me the effort of inputing a graph but would also facilitate testing by round-tripped from polyhedon to graph and back to polyhedron.

Polyhedron to graph

My OpenSCAD library developed for the Conway operators (used in the polyhedron site) has a function to place a polyhedron on its largest face, used for orientation for 3D printing. I played with functions to map each 3D vertex onto this face in the x-y plane so that a planar graph was constructed. This function works:

function vertices_to_graph(vertices,k=1.2) = [for (p=vertices) let (sph= xyz_to_spherical(p), r=sph[0], theta=sph[1], phi=sph[2], rp= r*pow(1-cos(theta),k)) [rp*cos(phi) , rp*sin(phi)] ];

where

function spherical_to_xyz(r,theta,phi) = //r is radial distance, theta is polar angle, phi is azimuth [ r * sin(theta) * cos(phi), r * sin(theta) * sin(phi), r * cos(theta)];

and the parameter k allows adjustment of the graph to ensure that it is planar. The edges are given by the original polyheral faces.

Some examples of the Platonic solids:

Here k=1.2 fails, but increasing k to 2.3 yields

which is still not satisfactory as triangles are lost in the middle.

Finally the Herschel solid:

There is more work to do here to get a balanced graph in the general case.

Graph to Polyhedron

The mapping from the polyhedron vertices to the graph loses information so its inverse is not a function. However if we can generate a rough approximation to the vertices we can then use canonicalisation to create a polyhedron with planar faces.

This function uses a fixed value for the radial distance and the inverse of the vertex-to-graph mapping:

function graph_to_vertices(vertices,R,k) = [for (p=vertices) let (d=norm(p), phi= atan2(p.y,p.x), theta = acos(1-pow(d/R,1/k)), r=R*cos(theta)) spherical_to_xyz(r,theta,phi) ];

The faces are the same as the original polyhedron.

Applied to the generated Dodecahedron graph, this results in a rather squashed, non-planer object: (I’m sure this mapping could be improved too)

Canonicalisation

George Hart developed the idea of canonical polyhedra and an algorithm to canonicalise a polyhedron. In my Conway code, this is used for constructing the dual. We can also use it on our approximate polyhedron and at least for the Platonic solids, we get back to the original:

[Note that this is only nearly the same. 8 of the faces are not exactly planar]

Applied to the Herschel graph I get:

This is not the same as the original because it is now in canonical form, with non-square large faces.

But George was here first!

Looking again at George’s article, I noticed a familiar graph - the Herschel graph! He says:

It is interesting because it is the simplest polyhedron for which there is no Hamiltonian cycle (a round trip via edges which visits each vertex exactly once). What is more interesting is that from the net it is not at all apparent that the solid has 3-fold symmetry, but the canonical form brings it out immediately. (I was surprised !) I believe it is also the simplest polyhedron with an odd number of faces that each have the same number of edges.

His solution is given as a VRML file which I’ve converted and added to my index:

https://kitwallace.co.uk/3d/solid-index.xq?mode=solid&id=Hart-Herschel

As expected, this is the same canonical form.

So George pipped both my work here (which draws on his work) and Christian’s derivation by quite some years. My hero!

Christian’s geometric version does add something however. His version is precise and parameterised so that non-canonical vesions can also be constructed. My polyhedral index lacks the ability to handle parameterised descriptions but that would be a nice improvement to work on.

Non-planar graphs

It surprised me that round-tripping works even if the graph is non-planar. Actually this makes sense since the properties of the graph aren’t involved in the transformations.

Graph as starting point

Round-tripping from an existing polyhedron is all very well but what if we have to start from the graph. It would be good to find a way around hand-coding the graph cordinates and faces - that’s for another day.

OpenSCAD code

A version is in GitHub but also downloaded from the polyhedral index

To do

The plane() and canon() functions are failing with certain polyhedra

Simpler mapping based r=pow(phi/180,k) where k > 1 spreads out the lower points to achieve planarity (but some fail in canonicalisation as above)

example of direct graph input

References

George Hart , Calculating Canonical Polyhedra https://library.wolfram.com/infocenter/Articles/2012/

Branko Grünbaum, Graphs of polyhedra; polyhedra as graphs. https://digital.lib.washington.edu/researchworks/bitstream/handle/1773/2276/Graphs%20of%20polyhedra.pdf;sequence=1

#polyhedra #openscad

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matlabassignmenthelp-blog · 5 years

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Functions in Matlab - A guide from professional programmers

Matlab functions are used in several coding lines, which correlate a single variable with the other variables, and each outcome is related to a single input, which is used to generate an essential part of any programming language. The functions in Matlab environment are saved in a specific library file such as script files and much more. They can execute more than a single input argument, and it can return the value of more than a single output argument.

The functions in Matlab operate with the defined variables in the predefined workspace, which is called as ‘the local workspace.’ If you want any Matlab assignment help, you can take our expert’s help from Matlabassignmenthelp.com, along with the coding of the Matlab programs.

Kinds of Functions in Matlab

In Matlab programming, functions can be generated or defined with the use of the syntax:

Function [01,02,03……,0N] = myfun(i1,i2,.....iN)

In the above line, myfun is the name of the Matlab function that can take the input arguments from i1,i2,...iN, and return the output value 01,02,03...0N. For declaring any of the functions in Matlab, the above statement can be first-line. There are certain norms which require for the proper name of functions and save them:

The name of the functions that start with the alphabets and have certain characters, numbers, or underscore tends to be valid.

The function definition and the file names can be saved in the function file that matches with the first function name in a particular file.

We can store functions that contain commands and function definitions. The functions in Matlab must be represented at the last of the script file name, and it must not have a similar name as file function.

The end keyword can indicate the end of a function. This is needed when any of the file function has a nested function. There are various kinds of functions in Matlab, and these are as follows:

Anonymous Function:

This function is not saved in the program file; instead of this, it is related to the variables that have the data type as function_handle. It can be defined in one statement and can have several output and input arguments. The syntax of the function is:

Fun=@(argumentlist)expression

Example:

sum=@(a,b) a+b;

res= sum(3,2)

When we run this, it will produce the output as:

res=5

The anonymous function can be written without any input or with the number of inputs and outputs. If the anonymous functions in Matlab have no input, then we have to use empty parentheses to use an anonymous function, for example:

myfun=@(a,b) (a-b);

a=5

b=2

c=myfun(a,b)

Output: c=3

Local Function:

The first line containing a primary function, and it is visible to other files, and they can be called in any of the command lines. The other function which is available in the same file is called a local function. These kinds of functions in Matlab can not be seen in the main function or the parent function coded in the same file, and they are not called from the main command line. They are also called as sub-function. They can be used as subroutines in other programming languages. They can be coded in script files as they appear at the last line of the script codes.

Unction [avg, med] = mystats(b)

a=length(b);

avg= mymean(b,a);

med= mymedian(b,a);

End

Function a = mymean(v,n) //example of local function

a=sum(v)/n;

end

To define a script in name file integrationScript.m, that is used to compute the value of the area under the curve of y=sin(x)^3 from 0 to pi.

xmin = 0;

xmax = pi;

f = @myIntegrand;

a = integral(f,xmin,xmax)

function y = myIntegrand(x)

y = sin(x).^3;

end

The output will be:

y=0.6495

a= 1.333

Nested Function:

The functions in Matlab that are used within the parent functions or in another function are known as nested functions. These functions can modify or use the variables which are defined in other functions or the parent functions. They have the ability to access the workspace where they are already defined, and they can be defined in the function’s scope. There are some requirements that a nested function needs to follow:

The end statement is not needed for all the functions, but for nesting, every function requires the end statement.

The nested function can not be defined within any control statement such as switch case, if-else, and much more.

The name or a function handle can be used to call a nested function.

Function current

nestfun

Function nestfun

x=2;

end

Private Function:

These kinds of functions in Matlab are seen merely to a defined set of functions. They remain in the sub-functions and can be denoted by keyword “private.” These functions can be seen in the parent folder or in the functions which are directly above the subfolder of private. They can be used when we want to restrict the scope of various functions. A user can not call these functions from the outside folder of a parent or the command line.

Function priv

disp(“Hi”)

Alter the folder that has a private folder and alter the name of the file to ‘present’ as:

Function private

priv

Alter the folder to the other location and call that present function

Present

The output will be: “Hi”

Conclusion:

The functions in Matlab can be utilized in various programming. They can be used to form an essential part of any coding language. The Matlab functions can be available for any private function or any global function with the use of the global variable. They can be used to fulfill the demands of business or any other organization.

The information mentioned above can be used for Matlab programming. Still, if any of the students or the programmer find any issue related to the Matlab assignments then, they can take Matlab Assignment Help as our experts are available to you for 24*7 with the quality content. We provide all the assignments at a reasonable price and deliver it before the deadlines. Therefore, contact us and get the assignments from our professional programmers.

#Functions in Matlab #matlab assignment help #matlab online help #Matlab homework help

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matekon · 5 years

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The “best” definition for the sinus and cosinus functions

I use the following personal conventions:

● - Definitions - Propositions I assume are true

○ - Theorems – Propositions I deduce from the definitions

I also prefer $\tau$ which equals $2\pi$ as the circle ratio

_____

In mathematics, there is a common phenomenon: there can be multiple ways of defining the same mathematical object.

For example, here are 2 definitions for an isosceles triangle:

● 1) A triangle is isosceles if it has 2 sides of the same length.

● 2) A triangle is isosceles if it has 2 angles that are equal in measure

These 2 definitions are “equivalent” in the sense that a triangle would be isosceles according to the first definition if and only if it is isosceles according to the second definition. (If you are into analytical philosophy, specifically Frege, you might say these definitions express different “senses” but have the same “reference”.)

The case of the isosceles triangle is pretty simple, but in mathematics, there can be definitions for objects which are equivalent but where it isn’t trivial is the slightest.

Even though it is totally frequent for one mathematical object to have multiple definitions available, the way modern mathematics work (by axiomatisation), we have to choose one definition as a “starting point” and then deduce its equivalence with other definitions later on.

So, with our example of the isosceles triangle, we could either choose the first proposition as our definition and then, the second proposition would follow as a theorem, or we could just as well do the reverse.

But, is there a “starting definition” that is “better” than the others? From experience, I would say that, in the point of view of a “pure mathematician”, this is totally irrelevant and doesn’t matter. But, I do think that we can say some definitions are “better” than others if we allow ourselves to use didactic criteria to evaluate them.

In this article, I will be interested with the functions sinus and cosinus, for which I encountered many different definitions in my school years. In Section 1, I will present these definitions and say what I like and don’t like about them using a didactic approach. Then, in Section 2, I will introduce a definition of these functions that I personally think is the best one and I will show that it is “equivalent” with some of the definitions of Section 1.

__________

Section 1 - The usual definitions for sin and cos

At secondary school, I learned the following definition:

Definition 1 (by the triangle):

$\bullet$ Let $\triangle\ ABC$ be a right triangle where $\angle ABC$ is the right angle

Then we define $sin(\theta)=a/c$ and $cos(\theta)=b/c $.

The pros for this definition are the simplicity of the language and the fact that it is directly applicable to problems of geometry.

Among the cons, we have that this definition only makes senses for $\theta \in ]0,\tau/4[$ (let’s immediately work in radians). Also, I don’t think that this way of presenting sin and cos makes it obvious how to visualize the graphs of these functions. (It is possible to see the graphs but it requires us to be kind of clever.)

You might also say that this definition doesn’t immediately allows us to evaluate the functions for a given input. But I don’t think this is such a big problem and I will explain it soon later.

In CEGEP, I learned the definition involving power series:

Definition 2 (by the power series):

$\bullet \: sin(t) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}t^{2n+1} = t-\frac{t^3}{3!}+\frac{t^5}{5!}-...$

$ \bullet \:cos(t) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!}t^{2n} = 1-\frac{t^2}{2!}+\frac{t^4}{4!}-...$

This definition has the main advantage of allowing us to directly calculate the values of the functions. But the inconvenience that immediately comes with these types of definitions is that they make us say: “Where is this coming from? What is its utility?”.

The reality is that I think the human mind prefers to start of with definitions that make us directly see why the object in question is interesting and relevant. And then, for a function, we should find a way to “evaluate” it later on.

Let’s also note that this definition doesn’t make the shape of the graphs any more obvious.

In university, I got introduced to the following definition:

Definition 3 (by the differential equation):

$ \bullet \: sin (t) $ and $cos(t) $ are solutions of the differential equation $ f^{\prime\prime}(t) = -f(t) $

We first note that, because this equation has infinitely many solutions, we need further specifications to precisely define what $sin$ and $cos$ are.

This definition for me just mainly shows us a new interest of the sin and cos functions: there are extremely useful tools for solving differential equations. (This is one of the main motivations behind Fourier Analysis.)

This makes us do an important realization: maybe for didactic reasons, the definition we want to use for different mathematical objects depends on the context: For doing regular problems of geometry, Definition 1 for sin and cos is this best one to use. But for the theory of differential equations, then Definition 3 is more relevant.

I do think this argument is very important. But I also have a weak spot for definitions that are kind of more intuitive, more visual and just more “neutral” and “universal” I would say. I think all these criteria apply to the definition I will introduce in Section 2, which I will call “Definition 4 (by the circle)”. In fact, this definition is quite common, but I have never seen it being formalized the way I am about to do.

____________

Section 2 - The “best” definition for sin and cos

Let’s start with a simple question: “How do I describe a circle?”.

Algebraically, the simplest circle is the one of radius 1 centered at the origin. We define it this way:

$ \bullet \: S^1 = \{ (x,y) \in \mathbb{R}^2 | \: ||(x,y)||=1\} $

We see this way of defining a circle works the following way: We take as points $ (x,y) $ on the circle all the solutions to the equation $ x^2+y^2=1 $.

But what I would like to do instead is to describe the circle with a function, not an equation.

I want a function $f $ that outputs a point on the unit circle given a real number input.

$\bullet\:f:\mathbb{R}\to\mathbb{R}^2$ where $Im(f)=S^1$

We immediately see that $f$ is a vector function (has vectors as outputs). Because of that, it can be separated into 2 scalar functions (have real numbers as outputs).

$ f(t) = (x(t),y(t)) $ where $ x: \mathbb{R} \rightarrow \mathbb{R} $ and $ y: \mathbb{R} \rightarrow \mathbb{R} $

In case you didn’t guessed it, $x(t)$ will become $cos(t)$ and $y(t)$ will become $sin(t)$. I will continue to write them as $x(t)$ and $y(t)$ mainly because this notation makes their role clearer and because they aren’t fully defined yet.

Now, what I want to do is to fully define the functions $x(t)$ and $y(t)$. To do that, I will enumerate a list of properties that I want these 2 functions to have.

Because I said I wanted $f$ to output a point on the unit circle, that implies:

$ f(t) \in S^1 \iff ||f(t)||=1 \iff ||(x(t),y(t))||=1$

$\iff \sqrt{x^2(t)+y^2(t)}=1 \iff x^2(t)+y^2(t)=1 $

With this, I will state the first property to these functions, which is the first part of their Definition 4:

$ \bullet \: (1) \: x^2(t)+y^2(t)=1 $

This property isn’t enough. To illustrate this, let’s remark that the following function $f^{*} $ does obey the property $ (1) $ but isn’t the “nicest” function we could think of:

Basically, we see that $f^{*} $ isn’t “continuous”, because it occasionally “jumps”. But, let’s say I want $f$ to be a function that goes continuously around the circle.

In fact, I want $f $ to be something more specific: “parameterized by arc length”.

This means the following:

● Let $ g(t) $ be a curve in space (2d in this case). Then $ g(t) $ is parameterized by arc length if the length of the arc between $ g(t_0) $ and $g(t_1) $ (where $t_1 > t_0 $ ) is precisely $ t_1 – t_0$.

(In the case of the unit circle, how is this idea related to “radians”?).

I won’t go behind all the theory behind it. It is easy to google anyway. For our purposes, I need to know the theorem that says:

$ \circ \: f $ is “parametrized by arc length” $ \iff ||f’(t)||=1 $

From that I deduce:

$ ||f’(t)||=1 \iff ||(x’(t),y’(t))||=1 $

$\iff (x’(t))^2+(y’(t))^2=1 $

From this, we get the second part of Definition 4:

$ \bullet \: (2) \: (x’(t))^2+(y’(t))^2=1 $

$ f $ being “parameterized by arc length” implies that $ f $ is countinous, as we wanted.

The properties (1) and (2) largely define $ x(t)$ and $y(t)$. In fact, with just these 2 properties, we can show that $ x(t)$ and $y(t)$ must obey Definition 3 (by the differential equation). To prove this is a very fun mathematical exercise. Anyway, here’s my demonstration:

We know:

$ \bullet \: (1) \: x^2+y^2=1 $ $ \bullet \: (2) \: x’^2+y’^2=1 $

We want to show that $ y ^{\prime\prime} =-y $ (The proof for $ x ^{\prime\prime} =-x $ is analogous)

$ (1) \Rightarrow \frac{d}{dt}(x^2+y^2)= \frac{d}{dt}(1) $

$\Rightarrow 2xx’+2yy’=0 $

$\Rightarrow xx’=-yy’$

So, we have: $ \circ \: (A)\: xx’=-yy’\\ $

$ (2) \Rightarrow x^2( x’^2+y’^2)=x^2(1)$

$ \Rightarrow (xx’)^2+(xy’)^2=x^2 $

$\Rightarrow^{(A)} (-yy’)^2+(xy’)^2=x^2 $

$\Rightarrow y’^2(y^2+x^2)=x^2 $

$\Rightarrow^{(1)} y’^2(1)=x^2 \Rightarrow y’^2=x^2 $

Consequently, $ \circ \: (B)\: y’^2=x^2\\ $

$ (B)\Rightarrow \frac{d}{dt}(y’^2)= \frac{d}{dt}(x^2) $

$\Rightarrow 2y’y ^{\prime\prime} =2xx’$

$ \Rightarrow^{(A)} y’y ^{\prime\prime} =-yy’ $

$ \Rightarrow_{*} y ^{\prime\prime} =-y $

(I will call this final equation $ (E_y) $ and use it later)

$QED $

(The last implication with an asterisk below really needs a bit of justification. Because $ y’ $ can equal 0 for some inputs, we can’t just divide by it. But, there is a way to clean up the mess and make the deduction valid.)

As I said before, Definition 3 (by the differential equation) is incomplete in its formulation. So, it’s not because I was able to deduce it from $(1)$ and $(2)$ that these 2 properties are enough to define $ x(t) $ and $ y(t) $.

What I will do now is that I will try do deduce Definition 2 (by the power series). Trying to do so, it will show me what I have to add to $(1)$ and $(2)$ to make the Definition 4 (by the circle) complete.

What I know:

$ (1) \: x^2+y^2=1 $ $ (2) \: x’^2+y’^2=1 $

$ (A)\: xx’=-yy’\\ $ $ (B)\: y’^2=x^2\\ $

$ (E_x) \: x ^{\prime\prime} =-x $ $ (E_y) \: y ^{\prime\prime} =-y $

To deduce Definition 2, I would like to find the Maclaurin Series for $ x(t) $ and $ y(t) $:

$ x(t) = \sum_{n=0}^{\infty} \frac{x^{(n)}(0)}{n!}t^n $

$ y(t) = \sum_{n=0}^{\infty} \frac{y^{(n)}(0)}{n!}t^n $

So, I need to find $ x(0)$, $x’(0)$, $x ^{\prime\prime} (0)$, ... and $y(0)$, $y’(0)$, $y ^{\prime\prime} (0)$,...

$(1)$ and $(2)$ aren’t enough to find the Maclaurin Series. So, I will add the following defining property for $sin$ and $cos$ that is honestly only justifiable as a convention:

$ \bullet \: (3) \: f(0) = (x(0), y(0)) = (1,0) $

So we’ve essentially just chosen the starting point for our curve $ f$. It could just as easily have been $ (0,1)$ or $ (\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} ) $, which also are on the unit circle. This choice for $ f(0) $ is, I think, mainly justifiable as a way to make Definition 4 (by the circle) equivalent to Definition 1 (by the triangle).

Now, let’s use this equation in the quest of finding the Maclaurin Series of $x(t) $ and $ y(t)$ :

From $(3) \: x(0)=1 $ and $ (E_x) \: x ^{\prime\prime} =-x $, I deduce:

$ x^{(4k)}(0)=1 $ and $ x^{(4k+2)}(0)=-1 $ where $k \in \mathbb{N}$

From $ (3) \: y(0)=0 $ and $ (E_y) \: y ^{\prime\prime} =-y $, I deduce:

$ y^{(2k)}(0)=0 $ where $k \in \mathbb{N}$

We are halfway done. But, for the next step, we need to be kind of clever and use an old equation we proved earlier:

$ (B) \: (y’(t))^2= (x(t))^2$

$ \Rightarrow (y’(0)^2=(x(0))^2$

$ \Rightarrow^{(3)} (y’(0))^2=(1)^2 $

$ \Rightarrow y’(0) = \pm 1 $

Again, there is a choice to be made, and again, it is a matter of convention.

It can be shown that choosing $ y’(0)= 1$ will make the vector function $f$ go counterclockwise around the unit circle and that choosing $y’(0)= -1$ will make it go clockwise instead. Yes, we will choose the first option because of the convention of how we measure angles.

$ \bullet \: (4) \: y’(0) = 1 $

This will be the last property we add to Definition 4. Let’s see what we can do with it:

From $ (4) \: y’(0)=1 $ and $ (E_y) \: y ^{\prime\prime} =-y $, I deduce:

$ y^{(4k+1)}(0)=1 $ and $ y^{(4k+3)}(0)=-1 $ where $k \in \mathbb{N}$

From $ (4) \: y’(0)=1 $ and $ (2) \: x’^2+y’^2=1 $, I deduce:

$ (*) \: x’(0) = 0 $

Finally, from $ (*) \: x’(0)=0 $ and $ (E_x) \: x ^{\prime\prime} =-x $, I deduce:

$ x^{(2k+1)}(0)=0 $ where $k \in \mathbb{N}$

Putting all of this together, we finally get the MacLaurin Series:

$ x(t) = \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k)!}t^{2k} $

$ y(t) = \sum_{k=0}^{\infty} \frac{(-1)^{k}}{(2k+1)!}t^{2k+1} $

And so we’ve just proven that Definition 4 (we’ve just completed) is equivalent to Definition 2!

For clarity, let’s put all the parts of Definition 4 together and we will posture $x(t) = cos(t) $ and $y(t) = sin(t)$.

Definition 4 (by the circle):

$ \bullet \: sin: \mathbb{R} \rightarrow \mathbb{R} $ and $ cos: \mathbb{R} \rightarrow \mathbb{R} $, where

$ (1)\: cos^2(t)+sin^2(t)=1 $

$ (2)\: (\frac{d}{dt}cos(t))^2+ (\frac{d}{dt}sin(t))^2 =1 $

$ (3)\: sin(0) = 0 $ (which implies $ cos(0) =1) $

$ (4) \: ( \frac{d}{dt}sin(t))|_{t=0} = 1 $

We can also put it on words like that:

Definition 4 (by the circle):

$ \bullet \: f(t) = (cos(t),sin(t)) $ is a function from $\mathbb{R}$ to $\mathbb{R}^2$ where $ Im(f) = S^1$ (unit circle). Also, $ f$ is parameterized by arc length, starts on $ (1,0) $ and goes counterclockwise.

The language can really be seeing as harsh, but once this definition is really understood, it allows us to directly visualize what $sin(t)$ and $cos(t)$ mean. It is also not hard to see the shape of their graphs, especially with the help to the following gif: http://i.imgur.com/jvzRYnC.gif

This is the reason why I think this is the best definition in a didactic point of view.

If we want a more accessible language for people who aren’t specialized in math, this formulation would also be valid:

Definition 4: ● If I start on the unit circle at $ (1,0)$ and I walk t units of distance counterclockwise while staying on the unit circle, my x position will be $cos(t) $ and my y position will be $ sin(t)$

I let to you the proof that Definition 4 is equivalent to Definition 1 (for $ t \in ]0,\tau/4[) $. It is definitely the most straightforward proof on the bunch.

#mathematics #trigonometry #differential equations #calculus

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testblogplzignore · 3 years

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Fuck it doing it by hand so I can see it better. This is light years slower than just writing it down but I get the advantage of having it on the computer

$ u(x) = \frac{T_2 - T_1}{h} x + T_1 $

$ w(x,0) = f(x) - u(x) $

$ f(x) = \delta ( h) $

$ w(x,0) = \delta (h) - \big( \frac{T_2 - T_1}{h} x + T_1 \big) $

$ w(x,0) = \delta (h) + \frac{T_1 - T_2}{h} x - T_1 $

$ w(x,0) = \sum_{n=1}^{\infty} c_n sin \big( \frac{n \pi x}{h} \big) $

$ \delta (h) + \frac{T_1 - T_2}{h} x - T_1 = \sum_{n=1}^{\infty} c_n sin \big( \frac{n \pi x}{h} \big) $

$ c_n = \frac{2}{h} \int_{0}^{h}\Big[ \delta (h) + \frac{T_1 - T_2}{h} x - T_1 \Big] sin \big( \frac{n \pi x}{h} \big)dx $

$ c_n = \frac{2}{h} \Big[ \int_{0}^{h}\ \delta (h) sin \big( \frac{n \pi x}{h} \big)dx + \int_{0}^{h}\ \frac{T_1 - T_2}{h} x sin \big( \frac{n \pi x}{h} \big)dx - \int_{0}^{h}\ T_1 sin \big( \frac{n \pi x}{h} \big)dx \Big] $

$ c_n = \frac{2}{h} \Big[ \int_{0}^{h}\ \delta (h) sin \big( \frac{n \pi x}{h} \big)dx + \frac{T_1 - T_2}{h} \int_{0}^{h}\ x sin \big( \frac{n \pi x}{h} \big)dx - T_1 \int_{0}^{h}\ sin \big( \frac{n \pi x}{h} \big)dx \Big] $

$ \int_{0}^{h}\ \delta (h) sin \big( \frac{n \pi x}{h} \big)dx = 0 $

with a couple steps

huh

why does the dirac delta always disappear as an initial condition

actually this makes sense because you want $ w(x,0) = -u(x)$ transient solution only so when $ T(x,0) = w(x,0) + u(x) $ you get

$ T(x,0) = f(x) -u(x) + u(x) = f(x) $

$ \frac{T_1 - T_2}{h} \int_{0}^{h}\ x sin \big( \frac{n \pi x}{h} \big)dx = \frac{T_1 - T_2}{h} \frac{ h^2}{n^2 \pi^2} \Big[ sin \big( \frac{n \pi x}{h} \big) - \frac{ h}{n \pi} x cos \big( \frac{n \pi x}{h} \big) \Big]_0^h $

$ \frac{T_1 - T_2}{h} \int_{0}^{h}\ x sin \big( \frac{n \pi x}{h} \big)dx = \frac{T_1 - T_2}{h} \frac{ h^2}{n^2 \pi^2} \Big[ - \frac{ h^2}{n \pi} \Big] $

$ \frac{T_1 - T_2}{h} \int_{0}^{h}\ x sin \big( \frac{n \pi x}{h} \big)dx = (T_2 - T_1) \frac{h^3}{n^3 \pi^3} $

$ T_1 \int_{0}^{h}\ sin \big( \frac{n \pi x}{h} \big)dx = - T_1 \frac{h}{n \pi } $

$ c_n = \frac{2}{h} \Big[ 0 + (T_2 - T_1) \frac{h^3}{n^3 \pi^3} - - T_1 \frac{h}{n \pi } \Big] $

$ c_n = \frac{2}{h} \Big[ (T_2 - T_1) \frac{h^3}{n^3 \pi^3} +T_1 \frac{h}{n \pi } \Big] $

I ain’t seeing how that cancels at t=0 maybe with the sin?

it should solve the boundary values

but

$ T(x,t) = w(x,t) + u(x) $

$ T(x,t) = \sum_{n=1}^{\infty} c_n sin \big( \frac{n \pi x}{h} \big) e^{\beta \Big( \frac{n \pi}{h} \Big)^2 t} + \frac{T_2-T_1}{h} x + T_1 $

$ T(x,t) = \sum_{n=1}^{\infty} \frac{2}{h} \Big[ (T_2 - T_1) \frac{h^3}{n^3 \pi^3} +T_1 \frac{h}{n \pi } \Big] sin \big( \frac{n \pi x}{h} \big) e^{\beta \Big( \frac{n \pi}{h} \Big)^2 t} + \frac{T_2-T_1}{h} x + T_1 $

$ n=1$

$ T(x,t)_{n=1} = \frac{2}{h} \Big[ (T_2 - T_1) \frac{h^3}{1^3 \pi^3} +T_1 \frac{h}{1 \pi } \Big] sin \big( \frac{1 \pi x}{h} \big) e^{\beta \Big( \frac{1 \pi}{h} \Big)^2 t} + \frac{T_2-T_1}{h} x + T_1 $

$ T(x,t)_{n=1} = \frac{2}{h} \Big[ (T_2 - T_1) \frac{h^3}{ \pi^3} +T_1 \frac{h}{ \pi } \Big] sin \big( \frac{ \pi x}{h} \big) e^{\beta \Big( \frac{ \pi}{h} \Big)^2 t} + \frac{T_2-T_1}{h} x + T_1 $

$ T(x,0)_{n=1} = \frac{2}{h} \Big[ (T_2 - T_1) \frac{h^3}{ \pi^3} +T_1 \frac{h}{ \pi } \Big] sin \big( \frac{ \pi x}{h} \big) + \frac{T_2-T_1}{h} x + T_1 $

try $ x=0.5 $ in a program as $ h > 0.5 $

k that didn’t go well

by thinking not like an idiot just check $ T_2 $ and see

What I don’t understand is why it’s not cancelling off

it should be

So the fallacy with this case is that it’s the first term in the series and other terms may cancel it off

the answer for $ T_2 $ is very close to just being $T_2$ which is what it’s supposed to be

small wiggle from $ T_2 $ with $ h=1$

at least it lines up with it at $ x=n $ or $ x=h$ if I didn’t set $ h=1$

What I’ve learned from this is the sinusoidal/exponential part to a scenario with fixed temperatures (I assume this is why it’s not working but I’m stupid) never has that part of the equation touch it. This is clearly to fix the temperatures. Gonna give up on this and find something with varying temperatures. It occurs to me that the reason it’s this simplified to fixed temperatures at the boundaries is because PDE’s are legitimately retarded to solve by hand as per one of the points to the class. Because of this I might not be able to derive an equation for a simplified scenario unless I try something different.

The coffee cup cooling assuming the temperatures at the ends don’t change looks like the reverse of this with the large wave spike coming from the cold side

I am apparently not applying my brain enough

I wanted math to show that this is the exact math for heat’s travel as you cool it down. Not that math is legible for the people who read my posts. In short I’ll still make a post about it and simply it largely down to because heat travels in waves there is an exponential term that dominates at very small times so for a small amount of time it travels very quickly assuming a temperature difference then slows as it gets cooler and cooler. I already knew this but I wanted to mathematically show it

I was also going to put a temperature on the delta term if it worked out. Like a physicist I am well aware it was supposed to be there

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scifigeneration · 7 years

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Pi pops up where you don't expect it

by Lorenzo Sadun

Pi is at the center of all circles. Holger Motzkau, CC BY-SA

Happy Pi Day, where we celebrate the world’s most famous number. The exact value of π=3.14159… has fascinated people since ancient times, and mathematicians have computed trillions of digits. But why do we care? Would it actually matter if somebody got the 11,137,423,895,285th digit wrong?

Probably not. The world would keep on turning (with a circumference of 2πr). What matters about π isn’t so much the actual value as the idea, and the fact that π seems to crop up in lots of unexpected places.

Let’s start with the expected places. If a circle has radius r, then the circumference is 2πr. So if a circle has radius of one foot, and you walk around the circle in one-foot steps, then it will take you 2π = 6.28319… steps to go all the way around. Six steps isn’t nearly enough, and after seven you will have overshot. And since the value of π is irrational, no multiple of the circumference will be an even number of steps. No matter how many times you take a one-foot step, you’ll never come back exactly to your starting point.

Calculating the area of a circle with wedges. Jim.belk

From the circumference of a circle we get the area. Cut a pizza into an even number of slices, alternately colored yellow and blue. Lay all the blue slices pointing up, and all the yellow slices pointing down. Since each color accounts for half the circumference of the circle, the result is approximately a strip of height r and width πr, or area πr2. The more slices we have, the better the approximation is, so the exact area must be exactly πr2.

Pi in other places

You don’t just get π in circular motion. You get π in any oscillation. When a mass bobs on a spring, or a pendulum swings back and forth, the position behaves just like one coordinate of a particle going around a circle.

youtube

Simple harmonic motion is another view of circular motion.

If your maximum displacement is one meter and your maximum speed is one meter/second, it’s just like going around a circle of radius one meter at one meter/second, and your period of oscillation will be exactly 2π seconds.

The area of the space under the normal-distribution curve is the square root of pi. Autopilot, CC BY-SA

Pi also crops up in probability. The function f(x)=e-x², where e=2.71828… is Euler’s number, describes the most common probability distribution seen in the real world, governing everything from SAT scores to locations of darts thrown at a target. The area under this curve is exactly the square root of π.

How did π get into it?! The two-dimensional function f(x)f(y) stays the same if you rotate the coordinate axes. Round things relate to circles, and circles involve π.

Another place we see π is in the calendar. A normal 365-day year is just over 10,000,000π seconds. Does that have something to do with the Earth going around the sun in a nearly circular orbit? Actually, no. It’s just coincidence, thanks to our arbitrarily dividing each day into 24 hours, each hour into 60 minutes, and each minute into 60 seconds.

What’s not coincidence is how the length of the day varies with the seasons. If you plot the hours of daylight as a function of the date, starting at next week’s equinox, you get the same sine curve that describes the position of a pendulum or one coordinate of circular motion.

Advanced appearances of π

More examples of π come up in calculus, especially in infinite series like 1 - (1⁄3) + (1⁄5) - (1⁄7) + (1⁄9) + ⋯ = π/4 and 12 + (1⁄2)2 + (1⁄3)2 + (1⁄4)2 + (1⁄5)2 + ⋯ = π2/6 (The first comes from the Taylor series of the arctangent of 1, and the second from the Fourier series of a sawtooth function.)

Also from calculus comes Euler’s mysterious equation eiπ + 1 = 0 relating the five most important numbers in mathematics: 0, 1, i, π, and e, where i is the (imaginary!) square root of -1.

A graph of the exponential function y=e^x. Peter John Acklam, CC BY-SA

At first this looks like nonsense. How can you possibly take a number like e to an imaginary power?! Stay with me. The rate of change of the exponential function f(x)=ex is equal to the value of the function itself. To the left of the figure, where the function is small, it’s barely changing. To the right, where the function is big, it’s changing rapidly. Likewise, the rate of change of any function of the form f(x)=eax is proportional to eax.

The relationship between an angle, its sine, cosine and a circle. 345Kai, CC BY-SA

We can then define f(x)= eix to be a complex function whose rate of change is i times the function itself, and whose value at 0 is 1. This turns out to be a combination of the trigonometric functions that describe circular motion, namely cos(x) + i sin(x). Since going a distance π takes you halfway around the unit circle, cos(π)=-1 and sin(π)=0, so eiπ=-1.

Finally, some people prefer to work with τ=2π=6.28… instead of π. Since going a distance 2π takes you all the way around the circle, they would write that eiτ = +1. If you find that confusing, take a few months to think about it. Then you can celebrate June 28 by baking two pies.

Lorenzo Sadun is Professor of Mathematics at the University of Texas at Austin

This article was originally published on The Conversation.

#mathematics #pi day #piday2018 #pi #3.1415 #probability #circle #calculus

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maximuswolf · 4 years

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Derivative of a parametric via /r/calculus

Derivative of a parametric

So I'm trying to take the derivative of a parametric where

X=sin(t)

Y=csc(t).

I find dx/dt and dy/dt and divide and I get dy/dx is equal to -csc2 (x).

I'm looking for the slope of the tangent line at t=pi/2, so x=sin(pi/2)=1.

Dy/dx =-csc2 (1)=-1

So I would think that the slope of the tangent line at t=pi/2 is -1, right? But then when I go to double check using a graph calculator, I find that the equation doesn't have any x values past x=-1. That means the slope of the tangent line is nonexistent, because there are multiple tangent lines now. But the answer key says that the slope of the tangent is -1 after all. Where did I screw up?

Submitted February 28, 2021 at 09:29AM by redditard_redditard via reddit https://ift.tt/2O3zHvz

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btsinspirationtakesme · 7 years

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Prompt: “Unless you want me to fail math class by non attendance, I have a lecture to get to!”

Smuttish Namjoon prompt x Y/N

AN: The Namjoon thirst is very real right now, but then again- when is it ever not real?

“I still don’t understand.” You said, exasperated.

Namjoon looked at you like you grew two heads. “It’s really simple, Y/N. Just look at how the value of 3.14 is always equal to pi and then always make sure you divide by both sides.” Namjoon said, making the equation that you were trying to figure out the easiest one ever. Too bad for you, it wasn’t clicking in your head. You needed to pass these tests for a minor course in math that you had taken stupidly, but your brain clearly couldn’t keep up.

You also were going towards beyond the point of caring. Your boyfriend had offered up his brilliant brain to tutor you, and it was working at first but you were slowly getting distracted. After so much explaining that wasn’t going in, you admired Namjoon’s resilience to teach you. You were currently both sat at your desk, Namjoon getting excited at the math problem talking in what seemed like to you meloidic jibberish before turning to explain the solution to you.

“Okay, so here’s what we do. The question asks us to divide the number of yields per person to the distance they travelled…” You decided to focus on his voice instead, the velvety smooth voice that you could drown in. He had such a good way of teaching, but your brain was refusing to soak anything in apart from how good he looked with his pen in his mouth when he was frustrated with a problem himself, or how he kept flicking back to look into your eyes whilst he explained a problem.

“Y/N?” Namjoon asked. He saw your eyes darken, as you balanced the side of your head with your hands looking at him. You smiled.

“Go on.” You said, looking back to your textbook and he carried on. God, his voice was sinful explaining integars to you. Positively sinful.

You decided to have some fun yourself, moving closer to him and resting your chin on his shoulder.

You pressed your body against his and wrapped your hands around his bicep.

“Are you even listening Y/N?” Namjoon asked, catching on to how you certainly weren’t paying any attention now.

“Mhmmm…you’re so good at explaining.” You said, nipping his ear.

“Y/N, you need to pass.” Namjoon said, but you were having none of his objections. You started to trail a series of kisses down his neck.

“You’re being a really naughty girl, Y/N.” Namjoon warned darkly, his voice on edge as you found his soft spot.

“Mmm..punish me.” You said sweetly. Namjoon pushed your body back to the chair and warned you to stay put.

“You’re going to listen, no touching- okay?” He said, resting a hand on your knee.

“Not even your hands?” You asked in sadness.

“Not even hands.” He warned, eyes now focused on the math problem he was last at.

“Look, Y/N. Question 3.” He said, before diving into another explanation. You swore his voice became extra deep and velvety as he explained this one. Everything about him explaining was making you wet. Normally, Namjoon and you had no problem getting physical but because Namjoon was in his study mode he was resisting any touch.

You finally decided to let go and try to focus, soon enough understanding some of the concepts he was explaining. What you didn’t realise was how far Namjoon’s hand had travelled as you got more and more engrossed in the math problems. He was just about to go under the hem of your skirt when you realised his game. As soon as you got another question right, Namjoon moved his hand further up.

“One more question, baby. I know you can get this right.” He flashed you a brilliant dimpled smile followed by a wink. He was so close to touching you.

“It’s 7.34″. You said, working out the value of ‘x’ in just under ten seconds.

Namjoon took some time trying to figure out if you were right, but when he did he smiled and finally moved his talented hands all the way to where you most wanted them.

“Answer these in 10 seconds.” He said with a mischievous smile this time, and you were confused at first but as you started answering questions within 10 seconds Namjoon pressed down his long fingers inside you causing you to clench your legs in frustration.

“Fuck.” You said, breathing out.

“No, that’s not the answer.” Namjoon drew his fingers out but you held them there, begging him with your eyes to not let go.

“No, it’s 177.” You said quickly.

Namjoon smiled.

“That’s right babe.” He said smiling again in delight, enjoying just how much you were progressing from his little study technique.

“345.” You said, working out the last question in a hearbeat because you really wanted the prize that Namjoon was after.

“Babe, you’re a math slut.” He said with a chuckle as he closed the textbook and sped up his fingers.

You smiled in delight seeing as you were now done with studying.

“I don’t care, as long as you do this to me, I’ll change my major.” You said in desperation as Namjoon basked in how needy you were.

“Mmm, I should come over and tutor you more often.” Namjoon sighed, enjoying the sight of you writhing.

“Fuck, fuck, fuck!” You yelped as he sped up his fingers inside of you.

You were about to come to a high, an expected mind shattering orgasm induced by your fuck-as hot boyfriend who sounded like pure sin even when he was doing his best to tutor you when Namjoon suddenly withdrew his hands.

“Mmmm…” He said licking you from his fingers.

“What the hell, Namjoon!” You yelled in frustration.

“Babe, you got 2 questions wrong. I wanted to see how riled up you could get.” He said, a shy smile on his face.

“But I can still pass with 7/10!” You protested, flustered.

“Yes, but I’m your tutor. You’re not walking out of that exam unless it’s 10/10.”

You huffed in annoyance. “Did I do this to you when you wanted Biology tuition?” You asked. From what you remember, you put more work preparing to go through biology with him even though he was doing better than you in biology.

“No, you wore those shorts and I got a boner everytime you leaned in.” Namjoon said. “This is payback.”

You got up and groaned. “This is why we shouldn’t tutor each other.” You said.

“It’s not my fault you start gushing everytime I talk about the value of an integar.” Namjoon smirked as he teased you.

“Yah! Get out of my sight Kim Namjoon.” You protested.

Namjoon laughed throwing his head back. “I love you babe, do you also get this wet when I’m on stage?” He asked, pulling you towards him as he lovingly doted his eyes on you and rested his hands on your waist.

It was impossible to look at anything but him by the way he closed in on you.

“I’ll let you find out next time.” You said winking, before kissing his lips briefly and breaking out of his hold.

“Unless you want me to fail math class by non attendance, I have a lecture to get to!”

#namjoon #bts reactions #bts #bts scenarios #bts fanfic #bangtan

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mathematicianadda · 5 years

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Approximate recursion defined by alternating summation?

Problem & Question

Given a constant $k\ge 2$, we define the sequence $a_k(0)=0,a_k(1)=1$ and for $n\ge2$:

$$ a_k(n)=\sum_{i=1}^{\left[\frac{n}{k}\right]} (-1)^{n-i+1} a_k(i) $$

Where $\left[\frac{n}{k}\right]$ gives integer part of $\frac{n}{k}$. (The $\left[\space\right]$ represents truncation.)

For example, for $k=2$ we have the following scatterplot: [Click to see animated gif].

It appear as two interchanging curves repeat [following shape] in larger forms as we increase $n$.

Given $k$, can we find a real equation that would approximate "the curves" given by $a_k(n)$?

That is, the plot given by $a_k(n)$ can be defined as $\Gamma_1\cup\Gamma_2$ where $\Gamma_1,\Gamma_2$ are those "the curves".

Specifically, $\Gamma_1,\Gamma_2$ are graphs of two real functions $\gamma_1(x),\gamma_2(x)$ where $\gamma_1(x)=-\gamma_2(x)$ as they are symmetric over $y$-axis. The goal is to define them for $x\in\mathbb R$. Currently, you can say that $a_k(n)$ approximates $\gamma_1,\gamma_2$ at points $x=n\in\mathbb N$.

If it is not clear what I mean, clicking on the previous link "[following shape]", you can see the curve above the $y$-axis as $\Gamma_1$ given by $\gamma_1(n)$, and the curve below $y$-axis as $\Gamma_2$ given by $\gamma_2(n)$.

For simplicity, we can restrict ourselves to $k=2$ if needed.

My idea

The curves $\gamma_1(n)\approx-\gamma_2(n)$ generated by $a_k(n)$ remind me of a sine function whose period is being stretched by some function $g(n)$, and whose amplitude is being increased by some function $h(n)\ne 0$.

Is it possible to find such $g,h$ such that the following "sine form":

$$f_k(x):=h_k(x)\sin\left(\pi g_k(x)\right)\approx \gamma_1(x)=-\gamma_2(x)\space ?$$

To answer my question? This needs to be defined for all real $x\gt 0$.

Notice the above $f_k(x)$ form would have roots (be zero) at $x=g_k^{-1}(m),m\in\mathbb N$.

That is, lets observe "near-zeros" of the sequence $a_k(n)$ - The points where the curves pass over the $y$-axis and and are closest to it. Here are first couple "near-zeros" for $k=2$:

$$a_2(n)\approx 0 \text{ at } n\approx 2,4.5,12.5,33.5,84.5,204.5,480.5,1102.5,2494.5,5568.5,\dots$$

Meaning that the $f_k$ (the $\gamma_1,\gamma_2$) should be zero (have a root) somewhere near these "near-zeros". In the context of my "sine form" $f$ mentioned above, and for $k=2$, this means that we want to find $g_2$ such that for $m=1,2,3,\dots$ we have:

$$ g_2^{-1}(m)=2,4.5,12.5,33.5,84.5,204.5,480.5,1102.5,2494.5,5568.5,\dots $$

For example, here is the plot of $a(n)$ for $n\le215$ showing a near-zero around $n\approx204.5$, where we have $a(204)=-a(205)=-40\approx0$. (Hence I've taken $n\approx \frac{204+205}{2} = 204.5$)

But, the problem here is:

How to determine at which $n$ will $a_k(n)$ reach a "near-zero"?

The other problem that needs to be solved, is to find $h_k(n)$. For starters we need to know the growth of the absolute value of the sequence, $|a_k(n)|$? Then, the question remains:

Can we interpolate (approximate) a "closed form" of $h_k(n)$, to get the "amplitude" of $f_k$?

Alternatively, is there a better $f_k$ form to search for, than my "sine form" $f_k$?

from Hot Weekly Questions - Mathematics Stack Exchange from Blogger https://ift.tt/2RcMt8O

#Blogger

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rebrobindoesmath · 7 years

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Study With Me: Line Integrals

Hey guys! I’m currently studying for the Mathematics Subject Test of the GRE, which I plan on taking in the fall. One of the ways I like to study is by explaining the material to someone else. I currently have weekends off from research, and since Saturdays are for the boys, it leaves Sundays for GRE preparation.

Because of this, every Sunday, I’ll explore a different undergraduate topic that could appear on the Mathematics Subject Test. This week: Line Integrals.

I’ll talk about the following:

What is a line integral?

How do you calculate a line integral?

An Example

As a brief note, this post contains LaTeX code and will be much easier to read when viewed directly on my blog, where the code will compile!

What is a line integral?

Let’s first recall what we already know about integrals. We’re used to integrating functions of one variable over an interval [a,b]. We can think of this as integrating over the path on the x-axis from a to b, and the value of the integral as giving the area bounded by the curve y=f(x) over the path [a,b].

But! We can also integrate over paths that aren’t just straight lines along the x-axis. The resulting integral is called a curve, contour, or path integral. Most commonly, it is known as a line integral.

In this post, I’ll be talking about line integrals with respect to arc length.

Before we get into it, I’d like to start by defining what it means for a curve to be smooth. A curve, C, with parameterization r(t) = <x(t), y(t)> is smooth if the derivative r’(t) is continuous and nonzero. Additionally, we can say C is piecewise smooth if it is composed of a finite number of smooth curves joined at consecutive endpoints. Basically, this means I have a bunch of curves $C_1, C_2, ..., C_n $ that are all individually smooth and Each $C_i$ has its endpoints connected to $C_{i-1}$ and $C_{i+1}$.

Back to line integrals. Suppose we have a function f(x,y) and a smooth curve, C, in the x-y plane. We want to think about breaking C into n tiny pieces of arc length $\Delta s_i$. For each of the tiny pieces of C, choose any point $P_i = (x_i, y_i)$ and then multiply $f(P_i) = f(x_i, y_i)$ by the length $\Delta s_i$. This process is fairly similar to how we define integration for the case where the path is a line on the x-axis. We want to sum up these multiplied terms for all n terms. If the value of that sum approaches a finite, limiting value as $n \rightarrow \infty$, then the result is the line integral of f along C with respect to arc length. Below is a comparison of the single variable case integrating over a path [a,b] on the x-axis (left) and the line integral with respect to arc length over the curve C (right).

Note the notation used for the line integral. If we’re integrating over a path C, we write C at the bottom of the integral.

What does this mean geometrically?

The value of this integral is the area of the region whose base is C and whose height above each (x,y) point is given by f(x,y).

How do we actually calculate the line integral?

First, parameterize C. That is, for a parameter t, find the equations x=x(t) and y=y(t) for $a \leq t \leq b$. We consider C to be directed, which means we’re saying that we trace C in a definite direction, which is called the positive direction. Basically, we’re saying that t runs from a to b, so A = (x(a), y(a)) is the initial point and B = (x(b), y(b)) is the final point.

Since we have $(ds)^2 = (dx)^2 + (dy)^2$ (think Pythagorean theorem), we can write:

$\frac{ds}{dt} = \pm \sqrt{( \frac{dx}{dt})^2+(\frac{dy}{dt})^2}$

which can then be rewritten as:

$\pm \sqrt{(x’(t))^2+(y’(t))^2}$.

We use the + sign if the parameter t increases in the positive direction on C and the - sign if t decreases in the positive direction on C.

So, we have:

$\int_C f \,ds = \int_{a}^{b} f(x(t), y(t))\frac{ds}{dt} \,dt$.

An Example

Determine the value of the line integral of the function f(x,y) = x + y^2 over the quarter-circle x^2 + y^2 = 4 in the first quadrant, from (2,0) to (0,2).

Solution below.

First, we should parameterize the curve C.

Our parameterization:

$x = 2\cos{t}$

$y = 2\sin{t}$

with $0 \leq t \leq \pi/2$.

Then we find $ds$.

$ds = \pm \sqrt{(x’(t))^2+(y’(t))^2}$

$ds = \pm \sqrt{(-2\sin{t})^2 + (2\cos{t})^2}$

Using the fact that $\sin^2{t}+\cos^2{t} = 1$,

$ds = \pm \sqrt{4} = \pm 2$

We use positive 2, since t increases as we traverse C in the positive direction.

So, we have:

$\int_C f \,ds = \int_{0}^{\pi/2}(2\cos{t}+4\sin^2{t})*2 \,dt$

I’ll leave the integration as an exercise, but the final answer is 2(2+$\pi$).

Thanks for reading!

As always, feel free to send a message or an ask with any questions, comments, or concerns. Stay positive, friend!

#mathblr #mathematics #math #studyblr #calculus #line integrals #how to #instructional #rebrobindoesGRE #GRE prep

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devphilamaths · 3 years

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Solve sec(pi/2+theta) | sec(pi/2 +x) | sec pi/2 + x formula, Find Exact value sec pi by 2 + x

#Solve sec(pi/2+theta)#sec(pi/2 +x)#sec pi/2 + x formula #Find Exact value sec pi by 2 + x #sec(pi/2+theta)#se #si #sec #sin #cos #ten #cot #math #nda #cds #pi/2 #180 #90 #1200 #classy #gdlmx #gd

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craigbrownphd-blog-blog · 6 years

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Some Irresistible Integrals, Computed Using Statistical Concepts

#ICYDK: Below are a few integrals that you won't find in textbooks. Solving them is a good exercise for college students with some advanced calculus training. We provide the solution, as well as a general framework to compute many similar integrals. Maybe this material should be part of the standard math curriculum. Here, p, q, r are positive real numbers, with q larger than p. The Gamma symbol represents the gamma function. It is possible that these results are published here for the first time. These are known as Frullani integrals, although the ones mentioned here are not covered by Frullani's theorem, nor any recent generalization that I am aware of. Indeed, AI-based automated integration platforms such as WolframAlpha can not find the exact value (only an approximation) while they are able to compute standard Frullani integrals exactly. My approach to derive the exact values is different from the classical approaches, as it relies on the statistical concept of expectation, possibly leading to interesting areas of research. How to compute such integrals? These integrals are a particular case of the following main result, proved in the next section: where g(x) / x tends to 1 as x tends to infinity, and f is a bounded function with a finite expectation. Some additional conditions may be required, for instance the fact that there is no singularity point in the above quotient, and that g(x) has a lower bound that is strictly positive. The expectation of f, also called average value, is defined as For instance, if f(x) = sin(SQRT(x)), then the expectation exists, and it is equal to E(f) = 2 / Pi. (Prove it!) The main result introduced at the beginning of this section, is rather intuitive but needs great care to prove it rigorously, including correctly stating the required assumptions on f and g to make it valid. Some cases might require working with non-Riemann integrals. Here we only provide the intuitive explanation. Proof of the main result (sketch) Here p, q and n are integers, with q greater than p. We are interested in the case where n tends to infinity. We approximate integrals using the Euler-Maclaurin summation formula. The approximations below become equalities as n tends to infinity. We used the classic approximation of the harmonic series to make the logarithm terms appear. Note that for large values of k, g(k) is asymptotically equal to k. This was one of the requirements for the formula to be valid. We also have: Using the change of variable y = x / q in the first integral, and y = x / p in the second integral, we obtain: Let us remark that: * q / g(qy) is asymptotically equivalent to 1 / y (for large values of y) * p / g(py) is asymptotically equivalent to 1 / y * both integrals diverge, so the impact of small values of y eventually vanishes in each integral separately * the difference between the two integrals converges In view of this, we have: This concludes the proof. Related Problems * Four Interesting Math Problems * Curious Mathematical Problem * Two Beautiful Mathematical Results - Part 2 * Two Beautiful Mathematical Results * Number Theory: Nice Generalization of the Waring Conjecture * Yet Another Interesting Math Problem - The Collatz Conjecture To not miss this type of content in the future, subscribe to our newsletter. For related articles from the same author, click here or visit www.VincentGranville.com. Follow me on on LinkedIn, or visit my old web page here. DSC Resources * Book and Resources for DSC Members * Comprehensive Repository of Data Science and ML Resources * Advanced Machine Learning with Basic Excel * Difference between ML, Data Science, AI, Deep Learning, and Statistics * Selected Business Analytics, Data Science and ML articles * Hire a Data Scientist | Search DSC | Find a Job * Post a Blog | Forum Questions https://goo.gl/RXZFDe

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top1course · 5 years

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How To Sell Millions Without Ever Opening Your Mouth! Copywriting Secrets Simple 7-Step Formula Pt.1

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Is it possible to sell millions of dollars without opening your mouth, well is he by the end of this, video I’m going to prove to you, yes is very, possible today we are going to talk about cooperating, secrets I have gave joining me, am I, cole director of social media, and gave manages all might, all my social media especially YouTube, so if you see that how I’m blowing up on YouTube how we went from, in the beginning of this year, with 100000 subscribers now almost the end, office year we’re going to hit a million Subs, subscribers, 10 x growth, is because, the gates help, okay so, he is going to act as almost as an audience or entering me on Sundays covered in question, i’ve been doing this for a long long time so sometimes the concepts, the preceptor I know that it’s, soul, so be with me for so long as I can conscious competent but maybe gave for him it’s a, Copper dick is a new thing for you right, is a new world, he will almost asked questions on your, behalf, now before I go into the, 7 step, on how to create, gray coffee, let’s just defy what cop rating, because I believe most people they could confuse what comprehend, you think copywriting, he’s about, writing, right and we hear I hear this a lot, odana I’m not a writer, my English is not, or I am not very, creative, what is not a, copywriting, what about closing, the way I Define carpeting is, closing in, print, okay closing in print or, closing the introvert right you can close using instead of using your, spoken words, you’re using your written words, imprint, to persuade convince and employing someone, want to make a Vine perch, do do a move to to do a make a decision whatever it might be what a goal that you might have, That’s all that all cooperating is, so if it’s closing you means it is about communication, it’s not about writing, right it’s not about, so don’t be intimidated by the whole concept of, writing cuz I flunked English twice when I was, in high school, but it doesn’t stop me from creating coffee, does May, millions and millions of dollars, deep Eddy questions, so I would say I see a lot in the YouTube comment where people they don’t really, noah copywriting is and how to, pi, and how, actually the origins of copywriting how, for any design or anything part of the business at all goes down to, word of mouth and being spoken and being read, correct, so how would someone I would, say how would they, Shift their thinking to start from a.

Perspective of their user, and the person you’re trying to reach through, i want you to think about cooperating good sometimes people might think it does that mean it is, that words are on the web page, it’s not that, every email that you sent out, is copywriting, are free, what pace are you create that’s corporate, baby social media post, cooperating, are free, even a video, write a video script a video, absl like a video sales letter, operating, infomercial that you, you see that’s copyrighted, so is everything that’s involved with, closing on a very massive, scale, on a mass scale to be. Copper, so you think about, a lot of entrepreneurs and business owners about that, you can hide copyright or you can also ask operator, but the biggest sin when it comes to businesses, you don’t have any understanding of what corporate is, Because that’s what most profit, this one sells appliance service, you can have exactly the same product, but the way that you write you communicate your tickling your marketing message, one, one message could produce $10,000, the exact same, product with a different message, you could create a million-dollar, i always think of this I will let me give you a metaphor, think of it as a, us, dollar bill, okay a dollar bill, the difference between a $1 bill, and $100 bill value, is the message that’s on, the piece of paper, well I want you to do in life, just why sometimes when you understand copywriting I’m not saying you have to be a master copywriter but by having some basic understanding, you can make some small changes may be the subject line, maybe the headline maybe to call to action, with a c LaMotta effort to see my money the same amount of traffic, Southern me now you getting to three times more results, we see this a lot of social media with everything, we were the same thing but we change the message we change the subject line, some minor tweaks, dollywood getting way more results right I wouldn’t be scared to, testing because we do that a lot. Encourages 100% and don’t be afraid to test, 100% guys test, and it is not about, the eagle where I know what works best if you don’t know I still don’t know, right just because I’ve experienced but sometimes you also make certain assumptions about certain things, and that’s a great thing now about the internet, doctor when I was doing carburetor, what do we like to recommend, right licking the envelope putting the stand, so we will send all this say, to a list, a5000 Direct Mail, Letters, good old like paper and print, and then 5002 this list, 5002 Dallas, and we’ll see what kind of response we get, it would take like weeks and months before we know the results, but now it’s internet, with tracking with software, you know exactly what’s happening so much, easier, so, pod up, being a good, copyright, it’s snowing., you and I will not the marketing genius, customer, let them, how you, if the offer, one good thing that I’ve learned from you see who is sometimes we look at what, even you guys Post in the comment and will reverse-engineer that, intoarce, girls page into, if you are not so sure about how how do I get started with this Incorporated, into my business, i’ll give you a very powerful but simple strategy, set up a time to talk to some of your best, okay and you actually talk to them, You can get them a little smoke if it’s a.

I’m just doing some research about my company, i want to do some marketing can pay you your my best and most loyal customer, can I have, 30 minutes, and if they do you can get them some morphe, certificate, starbucks whatever it is doesn’t matter what it is, and you talk, and you ask him questions like hey, you look before you, you do business with, company, what was your problem, right why did you choose us, and why do you stay with, they would tell you all these things, and which one of you want to record it ask for the commission you record it, and this is so easy, you take what they say, and you turn it into, coffee, so let’s say, give me example give me an example of something that people, alternate so let’s say, Yeah like, i can’t find any leads to my clients, and I’m really struggling finding leads and customers I go to these conference, and I talk to other people but none of them that really follow, oh okay so, leads all that so exactly that’s a pain that they have less a yellow product to solve their problem, easley, he’s the light. Could be a potential hotline, it could you want to call I get one of those, benefits team in a bullet point, how to quickly and easily generate more leads for your business without going to these, networking, boom right there you have, how to cook a medium talking about exactly what they say and it turned it into your copy, you do that for a number of customers song you are good, to go, see how easy this is not as complex as you think, right they would tell you your, Your customers they want to buy from you, but what do you want to know is the assurance that you know what, are going through, they want the shoes that, you’ll probably service can help himself., number one, look at 3 I want to teach you, 7 steps that you can take, every single time you buy coffee, you go through these steps, i Promise You by the end of the seven steps, you’ll coffee would be so much more powerful so much more compelling, . one and that is, identify your ideal, this is the most important part, because most people when did Duke operating I used to make this mistake, you so excited, you get you could tell it on the computer and you stopped typing, no you do not do that, you need to First understand truly understand, who your ideal customers, now that the way that is fine I do customers, Is three things, number one your ideal customer, they have a need, volume, so if I am a vegan, if you try to sell me steaks, we got a problem here right, you got to sell stick to a stick lighter so I should have a meeting at 1 for your part at this number one, number 2 is IHOP the ability, biopod, so I want it but also I could pay for, and number three if I have the authority, dubai Abaya service, so if, i am selling something to this a husband and wife, and I’m only talking to husband but we leased a wife that makes it, decision, that is not good, so it’s, they have their Nita want they have their building at the store if that’s the ideal customer, now once you know this, who do custom is even Facebook, This so much.

Targeting that we can do nowadays, with, he could be your interest it could be, gucci belong to it could be your age, ditto, hundreds of variations hundreds of, reference point hundred things you can do to Target, what you know exactly who you talkin bout that is, now you go, talk to them in your copy, one-on-one like this, and that’s what you do, i believe, good copy, 80% of research, and 20% lighting, so if Andrea Spinelli a month, sometimes sometimes 3 weeks a month, working on, coffee walk away campaign, you should spend like 2-3 weeks just in, before you write a single, you should have a very clear idea, exactly what the offer is, all these things, before you, i think another helpful tip, tipsy food that you gave the other day was, you really want to think about, who that ideal customer is in that, Customer profile and really Envision you talking to them one-on-one, like if you were just sitting down talking to them casual, don’t try to use very big mumbo jumbo, words just casual conversation as if, if you’re really trying, talk with them you want to talk to him on a one-on-one basis right you don’t want to be like right now I’m communicating with, with you, at any given time, your reader, audience, oil plus, they are, reading your coffee, the off, watching a video, by themselves, usual Toyota 101 connection, so I’m talking to you, i’m not talking like, speaking to a thousand a million people I’m not talking to groups, that public speaking is, completely different, what way comes to coffee, you want to be very personal I want to be like, one on one, assuming now you know your customer and I could tell you the amount of money you make is indirect, Proportion to how well you understand your car, i’ll say that one more time, the amount of money you make, it indirect proportion, 2 how well you understand, you know this business to fail, that, be entrepreneurs the business owner if they have this idea, on this widget I have this adventure time I am so excited about it, they go they take it to the market place nobody gets, because they don’t understand their, ideal, costume, it’s not enough to say hey my mom likes it my wife likes, did you say, when people typically say while everybody is my ideal cussed, i want to sell to every the white there is a bigamist, he kanaka even company ice Vegas Apple, even a couple days because she likes a Walmart that serves, everyone is my costume, no, yep Walmart, right you have so many your target, They all have different demographics so you know one could say that everyone is, especially for like small business are you mija medium-sized business you got to think you got to narrow it down, very much narrow down because the more you can narrow down the more personal, you can make it message, in Dominion, one of the best reactions you could get and I love this one man to Allen, he said let’s say you have an idea you can offer you want to test it out is that it is, you hang out with your target market, can you bring your offer and you show it to them, do you know what checkout is at 4, right, and if they say if they read the ad and they say to you, this is just as good as pretty good, your ex sucks, images no good, okay, i think if you have this out, No that’s no good.

Do you actually want is they read at, how can I buy some, is my credit card I want to buy some that’s when you know you got something, okay you need the real reaction because people vote with their wallet, don’t listen to Just what they say watch what they do they vote with a wallet, when did minute they put some money online okay I’ve got something, before that is all just like lips, so that’s number one research, knowing identifying do I do, step number, q and that is create an accessible offer, a compelling offer, this is probably the one of the most important steps Indian, higher formula, because you look at most businesses, the biggest challenges, they sound the same as everybody, the office not very compelling, oh you know what, this is my glock my competitors, And didn’t cuddle with you as well, maybe I’m a little bit better and maybe a little bit cheaper, what day is not, huge differentiator, so with what you do not think about how could you make, your offer as compelling as pause, i was talking with, one of a copywriter, and he was asking me well then do you believe, what is Montour, the audience, all the message, i say without a doubt, the market, i could have the greatest message, what I said the wrong people, what is distribution, we will just like yesterday last night we having dinner right, love’s like Rhett meet and all that doesn’t like Steve has so much, so I could have the great like lobster and crab and a great message this is awesome man, i don’t know it’s the right message, but wrong Market, play doesn’t work, so when it comes to offer you have to think about, What is it that your audience truly wants, right what is it that they did cleaning, what more what pain, what frustrations that they have they they just they’re sick and tired of that, so when you can come up with an offer this so irresistible, is easy to sell, in fact a great, offer, you can see, less, anyone still sell, you’ll need to use as many words, let me give you perfect, let’s say it’s a gate he has $100 bill, okay, can you give me $10 in exchange of $100, give me ten bucks I’ll give you $100 right now, that isn’t resistible, right there, what am I doing there is called selling money at a disco, so if you’re selling any kind of, business Improvement, offer or any offer that helps people to make money save money on investment, the concept, money saving money at a discount, Spandex with me, i’m going to teach you how to make, why, right, essay example people who teaches real estate investing, hey spend $5,000 on Discord, i’m going to teach you how to generate you know, 10000 a month in rental income, oh wow that’s a no-brainer, that’s only money at a discount, that’s an irresistible offer, maybe you’re not in the education space maybe you have, do some physical product how do you do this, in so many ways, maybe your irresistible offer is, a free 30-day trial, okay if you have seen those infomercials, where they have those are, skin care, the trifle 30 they see how you like it, example, or a strong guarantee, irresistible, add certain bonuses, jamaican resistant material quick story, the one time many years ago I was late night watching TV, that’s all this, eva Marcille just popped up my loved ones, And this this guy is demonstrating he has a pair of shoes on the desk and his.

Cutting it with a knife, and I’m like, this is cool, what is this is a was talking about house shopping Ibis, is it yeah that’s cool, okay I got the knife and then he cut actually a can, with with a knife, holy s*** this is cool, not going to use it but it’s cool right, so I was watching a commercial and then people like and then he has a lot of, people coming onto the TV talk with a testimonial, using this naive housewife, like people do any kind all kinds of, background that uses knife, and he said well you know what, can you buy today, you don’t just get one night, you get to knife, homelite holyfuck, this is awesome and then he goes on, this is the big knife, But what about All the Small Things, and then you get this knife, and then you get this knife, and then you get this knife, and then before you know what he’s talking about like, twenty f****** divinized, i don’t like this is crazy, this is awesome before the infomercial I don’t need a knife, i don’t need a knife, but offered it is so dramatic and so cool, and then he just dropped the mic and see if you order within the next 5 minutes, you don’t get, these 29th, we going to send you two sets of knives, that’s it I’m done right call number call immediately and by that is, list of offer it when did it the office so irresistible, logic goes out the window, outside bought to set up nice actually the lady cuz I’m like, what the f*** I need like 40 pairs of nice voices stupid I just dumb, It was at that moment it was so compelling, so think about how you could use that, maybe sometimes it’s, faster shipping Amazon that’s good Amazon, proserpine you get a fast, i don’t want to wait when I could also speak about how, you were saying how Amazon offers that fit did you pay them Accenture, that’s right cuz Amazon if you have Prime Membership Eva Prime member, hello, NC, that, with fine what they’re doing is essentially they’re charging you for membership fee, so they can buy more, applicants applying to get your product faster, so think about what Amazon does, and how they encourage you, a simple thing in your system, and it’s not like a hundred bucks or so right they raise the price people, put up a hole stink but everybody’s, bill pay the higher, i have seen people like us now, Amazon, hundred, housing people that you would think Prime is like, it’s a, luxury, everything nice and membership it’s not like a, necessity you’re paying, a hundred bucks you get some benefit in Amazon and, ended a video and entertainment, william painter sponsorship, so you’re paying them so I can buy more stuff, you got understand, i don’t know people, who are like, literally you would think like lower-income, the living in like, trailer park and stuff like that, neither would not think they would spend money on Amazon and yet their Prime members, it’s the last thing you think about, but to them, destify member, and I like you spend $100 a year, but to be a Prime member, is Solaris, it’s like a no-brainer, when you are a Prime member you thinking while you know, i’m going to pay for the shipping anyway, but I’m going to get my stuff faster, Plus you’re not getting all these other benefits I get Amazon need a video I can.

Music, and you may watch email watch it but you like the fact that, you have access, right, that’s irresistible,

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