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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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
21 views

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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70 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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57 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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44 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
51 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
20 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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73 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
55 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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23 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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11 views

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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19 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
60 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
56 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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28 views

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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34 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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30 views

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
23 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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37 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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13 views

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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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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44 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 ...
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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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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
136 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 ...
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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 ...
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1answer
68 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
34 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 = ...
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100 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
13 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)} = ...
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1answer
36 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, ...
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1answer
32 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 ...
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47 views

When is ${{x^2y} \over {(x^2+y^2)^\alpha}}$ continuous, using polar-coordinates

Given $$f ({x,y})= \begin{cases} {{x^2y} \over {(x^2+y^2)^\alpha}},&(x,y) \ne {(0,0)}\\ 0,&(x,y)={(0,0)} \end{cases}$$ For what values of $\alpha$, $f$ is continuous in ${(0,0)}$? I set ...
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1answer
87 views

Pullback metric, coordinate vector fields..

I'm doing this computation on $\mathbb{R}^3$ with cylindrical coordinates $(r, \theta, z)$, (which aren't defined on the whole of $\mathbb{R}^3$, but I don't care about that) and I seem to get a ...
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20 views

Find polar coordinates parameters given a cartesian point

I'm stuck with a rather simple problem: I have the following polar coordinates equation where I know the values for the $x$ and $y$ terms and I want to find $a$, $b$ and $\Phi$. $$x = a\cos(\Phi)$$ ...
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28 views

Direction angle of Line segment in polar coordinates

I have a line segment given by two points $A$ and $B$, that are $(r_1,\theta_1)$ and $(r_2,\theta_2)$ in Polar coordinates. I know that the direction angle of the line segment is given by: ...
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2answers
71 views

Expressing $(-8)^{\frac13}$ in polar form

I want to express $(-8)^{\frac{1}{3}}$ in polar and cartesian coordinates. What I did was to solve the equation $-8 = r^3e^{3i\theta}= r^3(\cos(3\theta)+i\sin(3\theta))$ which implies that I must ...
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12 views

Expressing a value in polar and cartesian coordinates

I have to express the value of $\sqrt{i}$ in polar and cartesian form. I really don't know where to start this problem, any hint could help me please!
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26 views

Relationship between Normal coordinates and Spherical Coordinates

I am using the following coordinates on $S^3: (\psi, \theta, \phi)$ where $$\begin{cases}x_0 = \sin\psi,\\ x_1 = \sin\psi \cos\theta,\\ x_2 = \sin\psi \sin\theta \cos\phi,\\ x_3 = \sin\psi ...
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3answers
26 views

ellipse polar co-ordinate conversion

I have a somewhat trivial question out of interest. Given the equation of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ why is the substitution $x = \sqrt{a}\cos t$ and $y = \sqrt{b}\sin t$ ...
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24 views

Divergence of outer product in polar coordinates

Right now I am trying to solve Euler's conservation equations for circular domain. Due to several factors, I am restricted to polar coordinates. I can't manage to correctly calculate divergence of ...
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4answers
59 views

How would I convert $2x^2 + y^2 + 3y = 0$ into polar form?

We are currently working with rectangular and polar equations. How would I convert $$2x^2 + y^2 + 3y = 0$$ into polar form? So far, I have tried to make the equation into rectangular and back ...
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2answers
27 views

How can I calculate, how the volume element transforms under change of co-ordinates?

Suppose I transform an integral $$I=\int f(x,y) \, dx \, dy$$ using polar coordinates, setting $x=r\cos\theta$ and $y=r\sin\theta$. We get $$ \begin{split} dx &= \cos\theta \, dr - r\sin\theta \, ...