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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How to convert $\theta = \pi/3$ into cartesian form?

How can I convert $$\theta = \frac{\pi}{3}$$ into cartesian form? What I get is $$ \theta = \frac{\pi}{3}\\ cos(\theta) = \frac{x}{r} = \frac{1}{2}\\ x = \frac{r}{2} $$ and I'm not sure what the ...
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1answer
37 views

Integrate $f(r,\theta) = r$ over the region between $r = a(1+cos\theta)$ and $r = a$

I am asked to integrate the function given in polar coordinates $f(r,\theta) = r$ over the region between $r = a(1+\cos\theta)$ and $r = a$. My answer is $$\int_{-\pi/2}^{\pi/2} \int_{0}^{a} r\cdot ...
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2answers
85 views

Calculate Coordinates on Arc, Based on Time of Day

Hopefully someone can help me out with this. I'm trying to calculate the position of a point on an arc, based on a percentage of distance along the circumference (% time of day). Sidenote - I'm ...
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1answer
27 views

Distance between two Polar-Coordinates

I choose two Points in Berlin with the coordinates: 1: lat: 52.511206 long: 13.546486 2: lat: 52.527501 long: 13.319206 With an online tool I got the ...
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1answer
37 views

Riemannian metric given in polar coordinates

the Riemannian metric of the euclidean plane is given in polar coordinates as \begin{align*} ds^2=dr^2+r^2d\theta^2. \end{align*} Consider more generally, \begin{align*} ds^2=dr^2+\psi(r)^2d\theta^2, ...
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1answer
13 views

Volume of a Cone using Cylindrical Coordinates

I'm aware of the usual method for calculating the volume by expressing the integrals for $dr$ and $dz$ in terms of $z$ to get the correct answer but when I attempted to solve it expressing everything ...
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0answers
19 views

Transforming points between two polar coordinate systems

I have 2 dimensional points (r, theta) defined in a polar coordinate system A, and a second polar coordinate system B with a known homogeneous transform T transforming between A and B in a Cartesian ...
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2answers
28 views

Integrate function $f(x,y) = x^2+y^2$ in circle $(x-a)^2+y^2 < a^2$

I am asked to integrate function $f(x,y) = x^2+y^2$ in circle $(x-a)^2+y^2 < a^2$ My answer is $\frac{3 \pi a^4}{2}$ but for some reason the answer of the textbook is $\frac{32 \pi a^4}{2}$. Does ...
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1answer
27 views

Integrate function $f(x,y) = y^2$ in $x^2+4y^2 \leq a^2$

I am asked to integrate function $f(x,y) = y^2$ in $x^2+4y^2 \leq a^2$ To do that using polar coordinates, how may I find the boundaries for $r$? Is there a procedure that always works (for ellipses ...
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3answers
52 views

Integrate $f(x,y) = e^{-x^2-y^2}$ in a circle with radius $1$ and center at $(0,0)$

I am asked to integrate $f(x,y) = e^{-x^2-y^2}$ in a circle with radius $1$ and center at $(0,0)$. The setup of the integral (in my solution) is $$ \int_{0}^{2\pi} \int_{0}^{1} e^{-r^2} r dr ...
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0answers
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Integrate $f(x,y) = \sqrt{4-x^2-y^2}$ inside circle with radius 2 and center $(2,0)$

An exercise asks to integrate $$f(x,y) = \sqrt{4-x^2-y^2}$$ inside circle with radius 2 and center $(2,0)$. When I set up the double integral I get $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} ...
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0answers
27 views

Converting region $\int_{0}^{2} \int_{0}^{x} f(x,y) dydx$ from rectangular form to polar form

How can I convert the region in $$ \int_{0}^{2} \int_{0}^{x} f(x,y) dydx $$ (which is basically a right triangle contained in the first quadrant) to a polar form? Im sure that $$0 \leq \theta \leq ...
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1answer
22 views

Why the degree change of the following polar form

From the following image: Why was $\theta$ changed to 3.08 instead of -3.12? Confused about that point. Details about the image: converting rectangular form to polar = $5^5$ ∠ $5 * -0.64$
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2answers
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Converting this complex number to polar form?

Given $ z = -1 - i$ ,I converted it to polar form, resulting r =$\sqrt 2$. And $\theta = \tan^{-1} (\frac{-1}{-1}$) = 0.785 rads, which seems incorrect with the solutions of my instructor I don't ...
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2answers
58 views

Converting $r=\sec^2(\theta)$ to Cartesian

I encountered this problem on my Calculus test today and am struggling to figure it out: Write $r = \sec^2(\theta)$ as a Cartesian equation. I have tried using all sorts of tricks on it ($x^2 + y^2 ...
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2answers
50 views

Find a Cartesian equation for the curve and identify it. $ r^2 \cos 2\theta = 1$

Find a Cartesian equation for the curve and identify it. $$ r^2 \cos 2\theta = 1$$ I'm confused by the $2\theta.$ I isolated $r^2$ to get $r^2 = \frac{1}{\cos2\theta}$ Now, normally if it was ...
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1answer
43 views

Dynamical System in Polar Coordinates

I have a dynamical system defined by : $ \dot x = {(x+iy)^n + (x-iy)^n \over2}$ and $\dot y = {(x+iy)^n - (x-iy)^n \over2i}$ Converting the system to polar coordinates gives the system: $\dot r = ...
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2answers
48 views

System of ODEs - from Cartesian to polar

Given the system of ODEs, $$\dot{x}=x^2+3y^2-1$$ $$\dot{y}=-2xy$$ How does one transform it into polar coordinates $(\rho, \theta)$? Here's my line of reasoning: let $x=\rho \cos(\theta)$, $y=\rho ...
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1answer
19 views

Rotating point by angle

Let $X = (c, 0)$. If I will rotate $X$ by, say, angle $\alpha = \frac{\pi}{4}$, how can I determine position of new angle? Will it just be $X' = (c + \cos\alpha, \sin\alpha)$?
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1answer
40 views

Convert polar velocity components to Cartesian

I haven't been able to find an answer to velocity component transformation from polar to Cartesian on here, so I'm hoping that someone might be able to answer this question for me. I am given a ...
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3answers
54 views

Proof of complex numbers

Let $w \in \mathbb C$ where $|w|=1$. I am trying to prove that there exists $ \theta \in \mathbb R$ such that $- \frac{i}{2}(w^n-w^{-n})=\sin(n\theta)$ for all $n\in \mathbb N$ To begin, I thought ...
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1answer
21 views

polar coordinates vector equation of a rectangle

We can write the equation of the circle in vector form in polar coordinates as: $$\vec{r}=R\hat{r}$$ ; where 'R' is the radius of the circle. Similarly, can we write the vector equation for a ...
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0answers
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How to define the domain of functions in polar coordiante?

I am rather confused by how we should assign the domain (the interval of values of $\theta$) in functions with polar coordinates. To be more specific, in the following image, we should find the ...
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2answers
54 views

Prove $dx*dy = r*dr*dφ$ using $d(r*cosφ)*d(r*sin(φ))$

I am trying to demonstrate that $dx*dy$ (in cartesian coordinates) is equal to $r*dr*dφ$ (polar coordinates). I know the image, but I want to follow an other way: $$x=r*cosφ$$ $$y=r*sinφ$$ ...
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1answer
12 views

Find the volume above and below a specified equation.

Find the volume above the x-y plane and below the surface $f(\theta, r) = \frac{5}{r+4}-\frac{5}{8}$. I do know how to find the answer using a double integration with respect to $r $ and $\theta$, ...
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0answers
18 views

Changing integral to polar coordinates

The problem is $$\iint_R \sqrt[2]{81-x^2-y^2}$$ and the conditions are $$\{(x,y)|x^2+y^2\le 81,x\ge0\} $$ So I figured if $x^2 + y^2 = r^2 \Rightarrow r^2 \le 81 \Rightarrow -9\le r \le 9$ for $r$. ...
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1answer
18 views

Finding the area of a region defined by a polar curve that is outside another polar curve region?

I'm stuck with a problem that despite a good bit of searching and even toying around with wolfram|alpha, I can't find an answer to: Find the area of the region inside r=5sinθ but outside r=4 I have ...
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2answers
47 views

Fourier Transform to solve Laplace's equation in cylindrical coordinates

I am trying to solve $\nabla^2 u = 0$ in cylindrical polar coordinates (and radial symmetry) $$ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{\partial^2 ...
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1answer
43 views

Find the Volume lying inside both the sphere $x^2+y^2+z^2=a^2$ and the cylinder $x^2+y^2=ax$

Taking the equation for the cylinder I completed the square to find $(x-\frac{a}{2})^2+y^2=\frac{a^2}{4}$ and the sphere clearly has radius $a$ and is centered at the origin. Now to solve this ...
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2answers
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proving the polar form of Laplacian operator [closed]

Prove $$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac{1}{r^2}\left(\frac{\partial^2 ...
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0answers
33 views

Bijection between polar and Cartesian coordinates

Let $(r,\theta)$ be the polar coordinates of a point in the plane. Then for any integer $k$, $(-r, \theta+(2k+1)\pi)$ and $(r, \theta+2k\pi)$ represent the same point. It seems intuitively obvious ...
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2answers
36 views

Finding area between polar curves using double integral

The question asks to find the area inside $r = 1 + \sin\theta$ and outside $r = 2 \sin\theta$ using double integrals. In my attempt, I found the intersection to be $\theta = \frac{\pi}{2}$. I ...
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0answers
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polar equation of a hyperbola - plus or minus?

I have a hyperbola with eccentricity of 5 and directrix with equation $r = −6 \csc(θ)$. I calculated it's polar equation to be $$r = \frac{30}{1-5\sin(\theta)}$$ Why wouldn't this be able to be ...
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1answer
20 views

Rewrite equation using cylindrical and spherical coordinates.

I want to rewrite the equation $z=x^2-y^2$ using cylindrical and spherical coordinates. The cartesian coordinates are of the form $(x,y,z)$. The spherical coordinates are of the form $(\rho, \theta, ...
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0answers
25 views

Surface Area and Volume of a Torus Using Polar Coordinates

Can the volume and surface area of a torus be derived using double integrals and a coordinate transformation to polar coordinates where $x = rcos(\theta)$ and $y = rsin(\theta)$? Equation for the ...
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0answers
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Find a region D from Cartesian coordinates to Polar coordinates

I'm trying to understand the relation between Cartesian coordinates and Polar coordinates. If I have a region $$D=\left\{(x,y): 0 \leq x \leq 2, \sqrt{2x-x^2} \leq y \leq \sqrt{4-x^2}\right\},$$ is it ...
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0answers
18 views

Finding polar coordinates from a point, why just $\pi$ in the one $y$ section

Why is it just one $\pi$ in the second one there?
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0answers
36 views

Christoffel symbols in polar coordinates calculation

I'm currently studying Riemannian Geometry and I would like to get familiar with the basic concepts. I considered the simple Riemannian manifold $(\mathbb{R}^2, can)$ with its Levi-Civita connection ...
3
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2answers
29 views

How to setup a double integral when the region is bounded by a circle and a parabola?

My task is this; Calculate$$\iint\limits_{A}y\:dA.$$ Where $A$ is the region in the $xy-$plane such that $x^2\leq y,\: x^2 + y^2 \leq 2$. My work so far: Our region $A$ is in the first and seccond ...
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0answers
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Area of infinitesimal polar rectangle

In taking a double integral in polar coordinates, I'm learning that we can break up the surface into little polar rectangles with $\Delta r$ and $\Delta \theta$. Therefore, the Riemann sum of a ...
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1answer
81 views

How to find the area inside the larger loop and outside the smaller loop of the limacon $r=\frac{1}{2} +\cos \theta$?

How to find the area inside the larger loop and outside the smaller loopof the limacon $r=\frac{1}{2} +\cos \theta$? Once the integrals are set up, I know how to solve them, but I'm having difficulty ...
2
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1answer
64 views

Work out the area enclosed?

I am doing a simple exercice and I think that either the book's solution is wrong or I misunderstood the problem. Here is the problem, 平面上で次の曲線又は直線で囲まれる図形の面積を求めよ。 極座標系について、曲線 $r = a(1+2\cos ...
2
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1answer
66 views

Using the Dirac delta function to find the density of point masses/charges

Here is an example from a textbook: Suppose there is a unit charge or unit mass at the point $(x,y,z)=(-1,\sqrt{3},-2)$; then in rectangular coordinates, the ...
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1answer
32 views

Converting from parts of a circle to polar coordinates

I have the area defined by $$A = \sqrt{x/4-(x/2)^2} < y < \sqrt{1-x^2)} \text{ and }0 < x < 1$$ and I'm supposed to find the integral of the function bound by these limits; $$I = ...
3
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2answers
82 views

Area of the region inside $r=\cos{\theta}$ but outside of $r=4\cos{3\theta}$.

I have been crazy finding the area of the region inside $r=\cos{\theta}$ but outside of $r=4\cos{3\theta}$. I can't decide the integral bounds
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1answer
21 views

Asymptotic behaviour of the riemannian metric in polar coordinates

I'm studying the section 7 ("Local Geometry in Constant Curvature) of chapter 5 of "Riemannian Geometry" written by Petersen. At the beginning there is a Lemma which says how behaves the metric $g$ ...
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1answer
12 views

What is the equation of a sphere of radius R centred at the origin in cylindrical coordinates?

I said that $r = R, -R \leq z \leq R$, and $0 \leq \theta \leq 2\pi$. Saying that $r = R$ is incorrect, however, but I don't understand why because clearly, at all points of the sphere the radius ...
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1answer
54 views

Can't get right solution for this polar problem…

Given: $$\frac{d\theta}{dt}=2$$ $$y = r(\sin\theta)=(3 \theta+\sin \theta)(\sin \theta)$$ Find $\dfrac{dy}{dt} = \dfrac{\left(\dfrac{dy}{d \theta}\right)}{\left( \dfrac{dt}{d \theta}\right)} = ...
3
votes
1answer
32 views

Confusion on polar coordinates of an ellipse

The polar coordinates of an ellipse are given by: $$x=\frac{abcos(\theta)}{\sqrt{b^2cos^2(\theta)+a^2sin^2(\theta)}}$$ $$y=\frac{absin(\theta)}{\sqrt{b^2cos^2(\theta)+a^2sin^2(\theta)}}$$ However, ...
0
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1answer
30 views

New limits when changing to polar coordinates for calculating a double integral

Calculate $$\iint_D {1 \over {(x^2+y^2)^2}} dxdy$$ when: $$D= \{\space (x,y) \in \mathbb R ^2 \space |\space {1 \over 2} \le x \space,\space x \le y \le \sqrt 3 x\space,\space x^2+y^2 \le 1\space ...