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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2answers
17 views

Determine an expression for the length of the curve $r = f(\theta)$ between $\theta = a$ and $\theta = b$.

Is my proof correct? We know that $$x=r\cos \theta=f(\theta)\cos \theta,$$ $$y=r\sin \theta=f(\theta)\sin \theta.$$ Taking the derivative and using the product rule, we have $$\frac{dx}{d\theta} = f'(\...
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1answer
15 views

n-dimensional integral of radial function

Suppose $f:\mathbb{R}^n\to\mathbb{R}$ only depends on the distance to $x$, $f(x_1,\ldots,x_n)=f(\sum_i x_i^2)$. I want to know the volume integral of $f$ over the region $\{x\in\mathbb{R}^n:x_i\ge 0, ...
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1answer
30 views

Converting the polar equation $r\cos2\theta = \frac1r+\cos\theta$ to Cartesian

Find the Cartesian equation of the polar curve $$r\cos2\theta = \frac1r+\cos\theta$$ Options: $x^2-y^2=1+x$ $x^2-y^2=1-x$ $x^2+y^2=1+x$ $x^2+y^2=1-x$ None of the above. I've solved both sides of the ...
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1answer
55 views

swap integral to polar

Trying to swap to polar and solve the following double integral, but I am not getting the same answer. $ \displaystyle\int_{-3/4}^{3/4} \int_{-\sqrt{3/4-x^2}}^{\sqrt{3/4-x^2}} 1/2\left(3-4\sqrt{x^2+y^...
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1answer
33 views

Convert $r=1+2\cos(2\theta)$ to Cartesian

I want to convert $r=1+2\cos 2\theta$ to Cartesian. $r=1+2(\cos^2\theta -\sin^2\theta)$ $r=1+2\left(\dfrac{x^2}{r^2}-\dfrac{y^2}{r^2}\right) \iff \dfrac{r-1}{2}=\dfrac{x^2-y^2}{x^2+y^2}$ $r$ won't ...
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3answers
53 views

Determine if $\lim_{(x,y) \to (0,0)} \frac{\sin \left(\sqrt{x^2+y^2}\right)}{\sqrt{x^2+y^2}}$ exists [closed]

Determine if $\lim_{(x,y) \to (0,0)} \frac{\sin \left(\sqrt{x^2+y^2}\right)}{\sqrt{x^2+y^2}}$ exists. I am having a very difficult time trying to solve this limit; can anyone help me solve and ...
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2answers
26 views

Naturalness of the cartesian coordinate system

The polar and spherical coordinate systems are intuitive when it comes to locating a point on a plane or in a space but they lack the naturalness of the cartesian coordinate system when it comes to ...
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1answer
46 views

How can I convert the polar equation $r^2=\ln(r^2\sin(\theta+\pi))$ into a cartesian equation? [closed]

If I have an equation like this, how can I convert it to a cartesian equation $$r^2=\ln(r^2\sin(\theta+\pi))$$
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Decomposing Circumferential strain to Radial and Tangential coordinates

I am currently working on a problem where a cylindrical sample is being compressed (say in the z direction, vertically) causing to increase in circumference (in x-y plane). I have a gauge which ...
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1answer
44 views

How do you write a contour in polar coordinates that covers all and only $z \in \{\Re(z) \in [c-a, c+a] \}$, but $ \Im(z) \in [-b,b] $?

I'm trying to do contour integration, but I need it over a specific subset of complex numbers $z$. Specifically, I need a contour that, for real numbers $a$, $b$, and $c$, it covers all and only $z \...
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Why $\int\sin\varphi\sqrt{r^2-2rz\cos\varphi+z^2}\,d\varphi$ results to having division by $rz$?

So I was trying to calculate this equation $\int \sin\varphi\sqrt{r^2-2rz\cos\varphi+z^2}\,d\varphi$, which I plug into calculator and get the result as $\frac{(r^2-2rz\cos\varphi+z^2)^{\frac{3}{2}}}{...
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1answer
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Showing $\frac{d^2y}{dx^2}=\frac{r^2d\theta^3 +2d\theta dr^2- rd^2 rd\theta}{(\cos\theta dr-r\sin\theta d\theta)^3}$

$\frac{d^2y}{dx^2}=\frac{r^2d\theta^3 +2d\theta dr^2- rd^2 rd\theta}{(\cos\theta dr-r\sin\theta d\theta)^3}$ given that $y=f(x)$, $x=r\cos\theta$, $y=r\sin\theta$ and $\theta$ is independent. $x=r\cos\...
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1answer
49 views

How can I Solve this integral by changing to another coordiantes?

How can we change to polar coordiantes and evalute this integral: $$ \int_{0}^\infty \int_{0}^\infty \left(x+y\right) ^{k} e^{-c(x+y)}d{x}d{y}$$ where $c$ is a positive constant, $k$ is a real ...
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Confused by choice of polar coordinates in stokes theorem problem

Say $F = [z^{3}, x^{3}, y^{3}]$ and circle is at $x = 2, y^{2} + z^{2} = 9$ So I'm trying to find: $$\int \int curlF \cdot n dA$$ So I understand that $C$ is the boundary curve of a disk and the ...
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0answers
17 views

Coverting to Polar Coordinates to Investigate Critical Points

In the second answer of the following link, a user converts to polar coordinates to investigate critical points. Can someone elaborate on this and explain why it works? I'm interested in what methods ...
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1answer
16 views

Using symmetry to find area enclosed by polar curve

I'm told that the area enclosed by the curve $r=a+5\sin\theta$, $a>5$ is given by $187\pi/2$, and I'm then asked to find the value of $a$. I know that the area enclosed by a polar curve is given by ...
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1answer
33 views

Converting the ellipse equation $r=\frac{a\left(1-e^{2}\right)}{1+e\cos\left(\theta\right)}$ from polar to canonical cartesian form

I have an ellipse with polar form equation: $$r=\frac{a\left(1-e^{2}\right)}{1+e\cos\left(\theta\right)}$$ where $e$ is eccentricity, and $a$ is semi-major axis. How do I convert this from polar form ...
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1answer
35 views

Partial Derivatives of polar co-ordinates

I am getting very confused when trying to find the partial derivative operators in polar co-ordinates. For example, I need to show that $\partial_{x}=\partial_{r}cos(\theta)-\frac{sin(\theta)}{r}\...
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50 views

Proving a circle is orthogonal to other circles

Let $K=(O,k), L=(A,r)$ be two circles. The radical axis of $K$ and $L$ is the line of the points $P$ that satisfy $$|OP|^2-k^2=|AP|^2-r^2.$$ Consider a third circle $M=(B,m)$. Let $S$ be the ...
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2answers
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Can I double integrate a function through r and theta instead of x and y?

Let's suppose a function $z=f(x, y)$ which has linear relationship with the distance from origin. For example we have a function $z=f(x, y)=max(-a \sqrt{x^2+y^2}+b,0)$, where $a,b>0$. And we're ...
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2answers
33 views

Proving $|PQ|^2$ is the sum of the powers of $P$ and $Q$ with respect to a circle

Reading a plane geometry book I found the following exercise: Given a circle $K=(O,k)$ and a point $P$, the power of $P$ with respect to $K$ is the quantity $|OP|^2-k^2$. Let $P$ and $Q$ be ...
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1answer
25 views

Graphing $(x^{2}+y^{2})^{3}=4x^{2}y^{2}$: when is cancelling a function, $r(\theta)$ in parametric equation valid

I am solving a question that asks me to graph $$(x^{2}+y^{2})^{3}=4x^{2}y^{2}.$$ I convert this into polar coordinates equation, to get $$(r^{2})^{3}=4(r \cos\theta)^{2}(r\sin\theta)^{2}\Rightarrow r^{...
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33 views

Compute an integral related with gaussian

Can anyone help me to solve this integral, I think we can try polar coordinate and some property of Gaussian density, but I stuck for long time. Also the WolframAlpha cannot compute this integral. $$ ...
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2answers
38 views

How to convert $\iint f_{xyz}(x,y,z)\,dy\,dz$ to polar coordinates?

It's been a while since I've done calc 3 work, and I'm a bit stuck on how to convert the integral $\iint f_{xyz}(x,y,z)\,dy\,dz$ into polar form. I would like to use $y = r \cos \theta$ and $z = r \...
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0answers
37 views

Cartesian to Polar coordinate systems: position vs vectors

I am having a bit of trouble wrapping my head around the difference between defining a cartesian vector in polar coordinates as oppose to defining a cartesian position in polar coordinates. Can ...
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2answers
42 views

Find $ \lim_{(x,y)\to(0,0)} \frac{\sin( |x| + |y|) + |y|(e^x - 1)} {|x| + |y|} $

$$ \lim_{(x,y)\to(0,0)} \frac{\sin( |x| + |y|) + |y|(e^x - 1)} {|x| + |y|} $$ I tried in this way $ \lim_{(x,y)\to(0,0)} \frac{\sin( |x| + |y|)}{|x| + |y|} + \frac{|y|(e^x - 1)} {|x| + |y|} $ the ...
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2answers
26 views

Cartesian velocity to polar velocity (Velocity Field Context)

I'm trying to derive the polar functions for the Velocity Potential Function $\Phi$ from its cartesian definitions of: \begin{align} \frac{dΦ}{dx} &= u \\\\\frac{dΦ}{dy} &= v \...
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0answers
28 views

Basis vectors and one-forms in curvilinear coordinates - normalized or not?

Here's my dilemma. We know, that $\partial_i$ forms a basis of vector fields and $\mathrm{d} x^i$ forms a basis of co-vector fields, so that we can write, in general (vector and covector) $$ v = v^i \...
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1answer
27 views

Converting into polar form for integration

So I am presented with a double integral which is can be evaluated easily enough as is. $$\int _0^{\frac{1}{2}}\int _{\sqrt{3}y}^{\sqrt{1-y^2}}xy^2 dxdy.$$ What I'm curious about is converting this ...
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1answer
14 views

Prove that the Cauchy-Riemann equations in polar coordinates are given in the following way: $ru_{r}=v_{\theta}$ ; $rv_{r}=-u_{\theta}$ [duplicate]

Let we have $f(z)=u(r,\theta)+iv(r,\theta)$ where $z=re^{i\theta}$ (being $r\neq{0}$). We have to prove that the Cauchy-Riemann equations in polar coordinates are defined in the following way: $\space$...
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0answers
26 views

Prove an equation is equal zero in phasor form (exponential)?

$$ Ve^{j\omega}\sum_{k=0}^{n-1} e^{jk (\frac{2\pi}{n}) } = 0$$ How Can I show that the equation will be zero? is it correct to say the exponential quantity will form a circle using Euler's Formula ...
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3answers
41 views

How do you convert this polar equation to Cartesian equation?

I have this equation to solve: $$r(1+\cos \theta) = 2$$ I know the answer is $y^2 = 4 - 4x$ but I don’t understand how to get there. I’ve tried multiplying both sides by $r$ as well as ...
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0answers
43 views

help me understand the transformation into polar coordinates from a differential-geometric view

I'm not entirely familiar with viewing differential equations from a differential geometry viewpoint. As far as I understand it, in terms of differential geometry, polar coordinates are a 2-...
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1answer
40 views

Where is my mistake in converting Cartesian to polar coordinates?

We always believe that $dxdy$ in cartesian coordinate is equivalent to $rdrd\phi$. So let's check: $x=r\cos\phi$ $y=r\sin\phi$. I differentiate the equations above to derive $dx$ and $dy$ $dx=dr\cos\...
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5answers
153 views

Messy Gaussian Integral

I am trying to understand how to better perform the following integral. $$\int^{\infty}_{0} x^4 e^{\frac{-x^2}{\beta^2}}\mathrm{d}x$$ I've done a little research and found that $e^{-x^2}$ doesn't ...
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3answers
67 views

Find the area between $r=1$ and $r=3\cos\theta$

Find the area between $r=1$ and $r=3\cos\theta$. I squared both sides to get $r^2 = 1$, then did $r^2(\cos^2 \theta + \sin^2 \theta) = (r \cos \theta)^2 + (r \sin \theta)^2$$ = x^2+y^2 = 1$ to get $x^...
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2answers
49 views

When do $A \ (0,0), B \ (\cos \alpha, \sin \alpha)$ and $C \ (\cos \beta, \sin \beta)$ form a right triangle? [closed]

Find the condition such that the points $A \ (0,0), B \ (\cos \alpha, \sin \alpha)$ and $C\ (\cos \beta, \sin \beta)$ form the vertices of a right triangle. I have trying this question but didn't ...
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1answer
21 views

In deriving the formula of curvature $K$ in parametric equations, why can $d\psi/d\theta$ be differentiated in terms of $\rho$?

In polar coordinates $\rho$ is the radius vector and $\theta$ is the angle the radius vector makes with the positive x axis. Let $\psi$ be the angle that the radius vector makes with the tangent to ...
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1answer
29 views

Finding bounds of integration non-graphically

I have this curve given in polar form: $$ \rho=4\sin2\phi $$ I'm trying to find the integration bounds of the area it creates, like this: $$ t = 2\phi \\ t \in [0 + 2k\pi, \pi+2k\pi], k\in Z \\ 0+2k\...
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1answer
34 views

Integrate curve given in polar coords solution verification

My solution to the below problem is $4\pi$ but the answer sheet says it's $8\pi$. Please verify my calculations: $$ \rho=4\sin2\phi \implies 4\sin2\phi \ge0 \implies \sin t\ge 0 $$ Where $t = 2\phi$ ...
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0answers
29 views

$2\text{D}$ Fourier Transform of Laplacian in polar coordinates

Consider a typical function written in standard $2\text{D}$ polar form: \begin{equation} f(\underline{r})=f(r,\theta)=\sum_{n=-\infty}^{\infty} f_n(r) e^{in\theta} \end{equation} executing the ...
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2answers
31 views

polar curves integration with $(dx)^2$?

Find the area which lies between the x-axis and the curve $x = sin(t)$, $y = sin(t)cos(t)$, where $0 \le t \le \pi/2$ I was able to sketch a graph in the x-y coordinate plane by making a table of $t$,...
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2answers
41 views

Locus of a point $P$ such that the tangents to 2 concentric circles issued from $P$ are orthogonal

If the tangent from a point P to the circle $x^2+y^2=1$ is perpendicular to the tangent from P to the circle $x^2+y^2=3$, then the locus of P is, So this is what the question means diagrammatically. ...
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0answers
21 views

Polar coordinates of a parametric curve

I have to find the polar coordinates of the following curve $$\alpha(t)=(2+\cos(t),2+\sin(t))$$ for $t\in[0,2\pi]$. Using $$x=r\cos\theta,\quad y=r\sin\theta$$ I found $$r=\sqrt{9+4(\cos(t)+\sin(t)}$$ ...
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1answer
21 views

evaluate limit with polar

In this limit i don't quite sure that the answer is correct can somebody please help me $$\lim_{(x,y)\to\ (0,0)} \frac{2x^2y+y^3}{x^2+y^2}=$$ $$\lim_{r\to\ 0} \frac{2r^2\cos^2\theta r\sin\theta+r^3\...
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1answer
33 views

Polar curvature

To find curvature in polar coordinates, is it the same formula as in rectangular but like with $dr/d\theta$ instead of $dy/dx$?
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1answer
32 views

Polar coordinates with arc-length instead of angle

Is it possible to define coordinates on the 2d cartesian plane with arc length and radius instead of angle and radius. For example I could have $$ \begin{split} s(x,y) &= \sqrt{x^2 + y^2 } \arctan(...
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2answers
42 views

How can I check that two lines are perpendicular and parallel in polar coordinates?

Given two lines $r\cos(\theta-\alpha_1)=k_1$ and $r\cos(\theta-\alpha_2)=k_2$, how can I prove that they are: Perpendicular $\iff$ $\sin\alpha_1\sin\alpha_2+\cos\alpha_1\cos\alpha_2=0$ Parallel $\iff$...
2
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1answer
32 views

Limit $\lim_{(x,y)\to\infty} e^{-e^{xy}}$ with polar coordinates

can i use polar to solve this limit? $$\lim_{(x,y)\to\infty} e^{-e^{xy}}=$$ $$\frac{1}{e^{e^{r^2\cos\theta\sin \theta}}}=$$ but i'm quite stuck here i think the denominator goes to infinity but should ...
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1answer
40 views

limit of multivariable function as x,y approach to infinity

can i solve this limit using polar coordinate? $$\lim_{(x,y)\to\infty} \frac{x^2+y^2}{x^2+(y-1)^2}=$$ $$\frac{r^2}{r^2-2r\sin\theta +1}=\frac{1}{1-\frac{2\sin\theta}{r}+\frac{1}{r^2}}=1$$

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