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Questions tagged [polar-coordinates]

Questions on polar coordinates, a coordinate system where points are represented by their distance from the origin ($r$) and the angle the line joining the point and the origin makes with the positive horizontal axis ($\theta$).

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Double integral of $xe^{-(x^2+y^2)}$

I have some troubles with the following double integral where D is $|x|\leq 1, |y|\leq 1$ $$ \iint_{D} xe^{-(x^2+y^2)} \,dx\,dy $$ I transform it to polar coordinates where $\theta~is [0,\pi /2]:$ $$ \...
TerribleStudent's user avatar
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2 answers
85 views

Area with double integral in polar coordinates

Determine the area interior to $y^2=2ax-x^2$ and exterior to $y^2=ax$. The area in artesian coordinates is $$\int_{0}^{a}\int_{\sqrt{ax}}^{\sqrt{2ax-x^2}} dydx$$. To convert it into polar coordinates ...
a_i_r's user avatar
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Finding an upper bound for / computing a $d$-dimensional integral

I want to know if there is a clever way of computing this $d$-dimensional integral, or if and how it is possible to use polar coordinates? \begin{align} \int_{|m|\leq k_F} d^d m \left( ...
putti.123's user avatar
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Polar parametrization of hexagon-like shape

I know the polar equation describing an hexagon is $$r_{\text{hex}}(\theta)=\dfrac{\sqrt 3}{2}\csc\left[\theta-\dfrac{\pi}{3}\left(\left\lfloor\dfrac{3\theta}{\pi}\right\rfloor-1\right)\right],\qquad\...
Conreu's user avatar
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1 vote
1 answer
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Periodicity of Polar Curves

I am a math educator preparing a unit on the calculus of polar curves. This is my first time teaching this particular unit, so it was also the first time I noticed that the "periods" of ...
NC1208's user avatar
  • 47
-2 votes
0 answers
26 views

Number of parameters needed to find a point on $S^n$

Firstly, let me point out that the following argument can be easily extended to $S^n$ for every natural number $n$, so I will just focus on $S^1$. Consider the circumference $x^2+y^2=1$, centred at $O=...
Davide Masi's user avatar
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Vector space structure for curve linear coordinates

The space ${\Bbb R}^2$ is the space of all 2-tuples $(x_1,x_2)$, where $x_1,x_2 \in {\Bbb R}$. Vector space structure is introduced in ${\Bbb R}^2$ in a straightforward manner. We can visualise this ...
WhyNót's user avatar
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0 answers
40 views

A question about the definition of derivative in different coordinate systems

The definition of the derivative goes like this: If $x$ is an interior point of a set $E \subseteq {\Bbb R}^n$, then a function $f: {\Bbb R}^n \rightarrow {\Bbb R}^m$ is said to be differentiable at $...
WhyNót's user avatar
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2 votes
1 answer
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Equivalence Between Polar Coordinates and Cartesian Coordinates? [duplicate]

I am trying to better understand the relationship between Cartesian Coordinates and Polar Coordinates. Here is what I understand: When going from Cartesian Coordinates $(x, y)$ to Polar Coordinates $(...
wulasa's user avatar
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Lucas numbers in polar coordinates: why does this monstrosity of a graph happen?

The other day I've been bored and was messing around in Desmos. I've had a silly idea: what kind of spiral would I get if I put down Fibonacci or Lucas numbers in polar coordinates? I've plotted a ...
Stepanchicko's user avatar
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60 views

A functional that maps a differentiable function $g:[0,1]\to[0,1]$ to a closed loop in two dimensions

How do I describe a functional $f$ that maps a differentiable function $g:[0,1]\to[0,1]$ to a smooth $h: [0,1]\to[0,1]^2$ such that $h(0)=h(1)=g(0)$ with the remaining constraints described in English ...
fool's user avatar
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4 votes
2 answers
92 views

Integral of two independent Gaussian with different mean

I am trying to solve the following integral: $$ \int_{-\infty}^{\infty}\int_{-c\left\vert x\right\vert}^{c\left\vert x\right\vert} \frac{1}{2\pi\sigma^{2}}\, \exp\left(-\frac{\left[y - \theta\,\right]^...
Massimiliano Datres's user avatar
2 votes
2 answers
75 views

Polar coordinates not to prove that the limit goes to zero

I have to study the differentiability in $(0,0)$ of the function $f(x,y)=\frac{x|{y^k}|}{\sqrt{(x^2+y^2)^3}}$ for $(x,y)\neq(0,0)$ with $k\geq0$, and $f(0,0)=0$. By definition of differentiability, ...
selenio34's user avatar
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3 votes
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Integral in polar coordinates

I was wondering if I had set up this integral correctly. If anyone could help me, I would greatly appreciate it. I am available in case there are any unclear things! $$\iint_D x^2+y^2dxdy, \quad D=\{1\...
Pizza's user avatar
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how to write this region $D$ in relation to $r,\theta$ in this $\iint_Df(x,y)dxdy$ where $D=\{x^2+y^2 \le1,x+y\le 1\}$ and $D=\{x^2+y^2\le1,x+y\ge1\}$

I have attached two photos showing the integration bounds and I find it tricky how to express $r$ and $\theta$ in those two, if $x=r \cos{\theta}$ and $y=r\sin{\theta}$, so any help is very much ...
A Math Wonderer's user avatar
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Finding or constructing Archimedes spirals with/from parametric lengths

I'm using Desmos, and have already combed through this site not finding anything close to what I need, nor have the equations and modifications I have tried been of help. Desmos Trial by Combat I need ...
CryptoMynd's user avatar
1 vote
1 answer
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Is there a way to derive the Polar Curve Area Formula using Parametrics?

I just finished up Calc BC, and one formula that my teacher never really went into the derivation of was how the area of a polar curve is given by $A=\frac12\int_\alpha^\beta r^2d\theta$. One that I ...
Aidan Hyde's user avatar
1 vote
1 answer
53 views

Laplace equation in polar coordinates in $\mathbb{R}^n$

I am studying partial differential equations, and when I learn the fundamental solutions, the book uses polar coordinates in $\mathbb{R}^n$ and the Laplace operator on the unit sphere to get the ...
MashiroHanser's user avatar
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1 answer
25 views

Area of a cardioid and a circle

Given the Cardioid by $f(\varphi)=3-3\cos(\varphi)$ and the circle given by $g(\varphi)=-6\cos(\varphi)$. I have 2 questions regarding its areas: Why $\frac{1}{2}\int_{0}^{\pi}g(\varphi)^2d\varphi=9\...
MiguelCG's user avatar
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How to modify off-center circle in polar coordinates so that input angle has a linear relationship with angle on circle?

I have a circle translated horizontally in polar coordinates described by the equation: $$r\left(\theta\right)=d\cos(\theta)+\sqrt{r_{0}^{2}-d^{2}\sin^{2}(\theta)}$$ where $d$ is the horizontal ...
R. Toy's user avatar
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1 answer
32 views

Volume of shifted and modified cylinder using polar coordinates

I have a shifted and modified cylinder $x^2+y^2=4x$, bounded below by $z=0$ and above by $z=\sqrt{16-x^2-y^2}$. I want to find its volume. Completing the square and conversion to polar coordinates ...
mohd's user avatar
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I want to find the volume of a noncircular cylinder that lies inside r=1+$cos(\theta)$ and outside the circle r=1,and top of the cylinder lies on x=z.

First I think as since our $x=z$ then the volume of our right non-circular cylinder's volume is ,where R is the region we integrating and dA is the change in area, $$ \int\int_RxdA $$ Then ...
Elfryionnn's user avatar
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2 answers
64 views

Arc length of a polar curve $r = -8\cos(t)$

If I am being asked to find the arc length of the polar curve $r = -8\cos(t)$ when I use the integral formula it gives me $16 \pi$. But since this polar curve represents a circle with radius 4, should ...
Mathematican's user avatar
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1 answer
34 views

How to solve this PDE involving polar coordinates?

PDE Equation with Polar Co-ordinates I struggle with using polar coordinates. What method would I use here - is it just the Method of Characteristics? If so, what would be the way to proceed and how ...
mfaczz's user avatar
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1 vote
0 answers
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How can i find the average of the product of two functions in polar coordinates? [closed]

I have 2 equations in polar coordinates that are only dependent on $\theta$ and not r: $$ f=F(\theta)\\ g=g(\theta) $$ How can I find the product of $<f.g>$. That is the product of the 2 ...
aved akesnof's user avatar
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What type of spiral is that on the picture ? and what is the formula of such?

I have found some types of spirals, and when I analysed those I have found, they do not met the criteria to shape the draw desired. And a observation point, bacause I think spirograph its a wrong name ...
FrakTool's user avatar
1 vote
2 answers
41 views

Continuous petal count function

Determine the number of times $r=\sin(nθ),~n$ not necessarily an integer, on a polar coordinate system intersects with a circle centered on the origin with radius 1 for all real numbers. The rose ...
Jack Arturo Nelson's user avatar
3 votes
1 answer
137 views

Solving Laplace's equation on semi-annular domain

Solve the following boundary value problems for the Laplace equation on the semi-annular domain: $ 1 < x^2 + y^2 < 2, y > 0 $ $ u(x, y) = 0, x^2 + y^2 = 1, u(x, y) = 1, x^2 + y^2 = 2, u(x, 0) ...
Tomer's user avatar
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1 vote
0 answers
21 views

How do I set boundaries in a multiple definite integral equation?

I've been trying to solve these types of problems for a while now and still can't figure out how to set boundaries for each integral in a multiple integral equation. One of the questions is as follows:...
Ethan's user avatar
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-1 votes
1 answer
96 views

The length of the curve

Find the length of the curve: $$\theta = \frac{r}{2} \sqrt{r^2+2}+\ln \left(r+\sqrt{r^2+2}\right),\quad 0 \leq r \leq 2.$$ Is it possible to apply the formula for calculating the length of a curve in ...
Gleb Cloudy's user avatar
2 votes
1 answer
105 views

Clarifications on the solution of a double integral: $\iint_X\frac{x^2y}{x^2+y^2}dxdy$

Calculate the following double integral: $$\iint_X\frac{x^2y}{x^2+y^2}dxdy$$ where $X=\{(x,y)\in \Bbb R^2\colon 1\leq x^2+y^2\leq2x\}.$ Here my confusion arises. Looking at the integrand the polar ...
Sebastiano's user avatar
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0 votes
0 answers
29 views

Equation of a line segment (passing through origin) in polar form

I know the equation for a line (not passing through origin) is $r = r_0\cdot sec(\theta - \phi)$ But that equation just doesn't hold for line passing through origin. If we have another parameter (not $...
stayin' alive's user avatar
1 vote
2 answers
59 views

Average distance from a point on a circle to the y-axis.

This is a simple question, but I must be making some mistakes as I don't seem to get the answer in the book. Question: Determine the average distance from a point on $x^2+y^2 = 9$ to the $y$-axis. My ...
Teodoras Paura's user avatar
1 vote
0 answers
78 views

Change the double integral from cartresian to polar cordinates

I have to solve this following integral, $$\int_0^{\rm A} \int_{\sqrt{R_g^2 - x^2}}^{\sqrt{R^2 - x^2}+r_0} \arcsin{\left(\frac{R_{\rm g}}{\sqrt{x^2+y^2}}\right)} {\rm d}y {\rm d}x$$ Here ${\rm A}$, $...
coolname11's user avatar
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0 answers
63 views

Similarity Solutions for the 2D Heat Equation in Polar Coordinates (with radial symmetry)

I'm attempting to solve the 2D heat equation expressed in polar coordinates, where $ \frac{\partial u}{\partial \theta} = 0 $ due to radial symmetry. This simplifies the equation to $ D (u_{rr} + \...
Kamal Ahmadov's user avatar
0 votes
2 answers
68 views

Why is the area enclosed by a polar equation given by the definite integral of $\frac{1}{2}r^2$ with respect to $\theta$?

Suppose there is a polar equation $r = f(\theta)$. To find the area bound by the polar equation and the equations $\theta = \beta$ and $\theta = \alpha$ where $\beta > \alpha $, one must use $$\...
19360254735168's user avatar
2 votes
1 answer
60 views

Prove or disprove: limit of a polar graph is the unit disk

Let $r:[0,\Theta]\to [0,1]$ and $D\subset\mathbb{R}^2$ be the defined as follows: $$r(\theta)=\theta-\lfloor\theta\rfloor \\ D = \{(x,y)\in\mathbb{R^2}\ |\ x^2+y^2 \leq 1\}$$ Note that $D$ is the unit ...
Amit Zach's user avatar
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1 vote
1 answer
91 views

Finish this proof of gradient in polar coordinates

In my exercise, we want to derive the formula of the gradient in polar coordinates. We end up showing that $$\nabla f(r\cos \theta , r\sin\theta)=\frac{ \partial g }{ \partial r }(r,\theta)e_{r }+\...
KiwiKiwi's user avatar
  • 169
0 votes
1 answer
65 views

A question about a multivariable limit

I have $$\lim_{(x,y)\to (1,0)} \frac{xy^2 -y^2}{(x-1)^2+3y^4}$$ where the answer must be "the limit does not exist". I set $y=m(x-1)$ and get $$\lim_{x\to 1} \frac{m^2(x-1)}{1+3m^4(x-1)^2}=0....
Ninja's user avatar
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0 votes
0 answers
60 views

Deriving the plane wave scattering from a conducting half plate

Consider the two-dimensional problem of a plane wave of angular frequency $\omega$ propagating in the direction $\hat{\textbf{k}}=cos(\phi^{i})\hat{\textbf{x}}+sin(\phi^{i})\hat{\textbf{y}}$ impinging ...
Chris's user avatar
  • 469
1 vote
0 answers
50 views

Integrating the multivariate normal distribution over an ellipse

I have learned that if we have two real-valued random variables $X$ and $Y$ that follow the centered multivariate normal distribution, this means that there exists a $2 \times 2$ symmetric matrix $A$ ...
Polyjuice Potion's user avatar
3 votes
1 answer
72 views

Error in graphing the polar equation $r=-1+\cos(\theta)$. I get a different answer than the book for cos(30degrees). Am I wrong?

I have included a picture of the table that is shown in the answer key. Most of the answers coincide with what I get by using various calculators, but some don't and I don't know if I am making some ...
David A.'s user avatar
0 votes
0 answers
16 views

Law of the areas (2nd of Keplero) proof

I was reading the following proof of the law of the areas (the generalization for central forces): Consider a plane curve $t\mapsto (x(t),y(t))$, that in polar coordinates is given by $t\mapsto \rho(...
Luigi Traino's user avatar
-1 votes
1 answer
48 views

Differentiate $x=rcos(\theta)$ with respect to y

So we know: $x=rcos(\theta)$, $y=rsin(\theta)$ and $x^2 + y^2 = r^2$. I assume it will help to consider $r$ and $\theta$ as functions of $x$ and $y$, but I am not sure how to incorporate this.
PeakyBlaze7788's user avatar
0 votes
2 answers
67 views

Calculate $\lim_{(x,y)\to (0,0)}\sin(xy)\left(\frac{x^3-y^5}{(x^2+y^2)^2}\right)$ without polar coordinates

I have this limit in two variables: $$\lim_{(x,y)\to (0,0)}\sin(xy)\left(\frac{x^3-y^5}{(x^2+y^2)^2}\right)$$ Now I know that $(x+y)^2=x^2+2xy+y^2$ and $$|xy|\leq x^2+y^2\quad \text{Cauchy-Schwarz ...
Sebastiano's user avatar
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0 votes
0 answers
21 views

How to find limits of integration when converting to polar coordinates

I'm specifically struggling on finding the integration bounds for $\theta$ as usually the bounds for the radius are clear to me. For example, for the problem $\int\limits_{D} \log(x^2 + y^2) \, dA$ ...
Layla16's user avatar
  • 137
2 votes
0 answers
24 views

Negative $r$ in polar coordinate while integrating [duplicate]

The question asks $\iint_R (3x+4y^2)\; dA$ where $R$ is the region in the upper half plane bounded by the circles $x^2+y^2=1$ and $x^2+y^2=4$ $$\int_0^\pi \int_1^2 (3r\cos \theta + 3r^2 \sin^2 \theta) ...
IrbidMath's user avatar
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1 vote
2 answers
54 views

Intersection of two hyperbolas in polar coordinates.

I want to find the intersections of two hyperbolas in polar coordinates. One of their foci coincides, we use this as the pole. (The right focus of the left hyperbola is the same as the left focus of ...
hyperbolaintersection's user avatar
0 votes
1 answer
37 views

Volume below the cone $z=2\sqrt{x^2+y^2}$ for $x^2+y^2\leq4$

For $x^2+y^2=4$, $$z=2\sqrt{4}\Rightarrow z=4$$ Since the radius of the basis is $2$, then the volume of the cone is $$V=\frac{\pi\cdot2^2\cdot4}{3}\Rightarrow V=\frac{16\pi}{3}$$ However, using ...
mvfs314's user avatar
  • 2,084
0 votes
3 answers
70 views

Transforming partial derivatives to polar coordinates [closed]

I have to convert the following expression $V(x,y) = x\dfrac{\partial f}{\partial y} - y\dfrac{\partial f}{\partial x}$ to polar coordinates. How do i express the partial derivatives in terms of $r$ ...
kopec's user avatar
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