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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33 views

Area enclosed by a circle and leminscate

Find the area enclosed by a circle $r=4\sin\theta$ and out of $r^2=8\cos 2\theta$ I have tried the following integral $\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\int_{\sqrt{8\cos2\theta}}^{4\sin\theta}...
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1answer
24 views

Can't find the area of a polar region

I've ran into a bit of a stopper on this one problem. I solved this other problem like this yesterday but this one seems to cancel itself out to zero. I'm not sure what I'm doing wrong with this ...
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2answers
33 views

Finding a circle in polar coordinates

I have converted the system of ODEs, $$x'=x-y-x(x^2+5y^2)$$ $$y'=x+y-y(x^2+y^2),$$ to polar coordinates and got this: $$ r' = r-r^3(1+4\sin^2(\theta)\cos^2(\theta))$$ $$\theta'=1+4\cos(\theta)\sin^3(...
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2answers
72 views

Phase portrait of ODE in polar coordinates

Given the system of ODEs in polar coordinates, $$r' = r(1-r^2)(4-r^2)$$ $$\theta'=2-r^2,$$ one can determine its equilibrium points and limit cycles as follows: $\gamma_1:= \begin{cases} r = 0,\\ \...
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16 views

Can the limits be applied differently to the following multiple integration?

The question is to change the cartesian form to the corresponding polar form: $$\int_0^a\int_y^a{\frac{x^2\,dx\,dy}{\sqrt{x^2+y^2}}}$$ The limit when applied in the format $\theta =0$ and $\theta = \...
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1answer
19 views

Jacobian matrix for ellipsoid

ive been asked to fine the jacobian matrix for an ellipsoid $$x^2/a^2 + y^2/b^2 + z^2 / c^2 = 1$$ ive been looking online for the parametric equations and i get two different answers $$x=a\cos(u)\...
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2answers
35 views

Differentiation with polar coordinates

I'm sorry if this is supposed to be something basic but I'm not being able to understand if r is as given above, how have they worked out r dot? What have they differentiated the x,y and z coordinates ...
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2answers
24 views

Parabola equation from cartesian to polar representation

I've got the following equation: 0) $ \frac{(y-y_p)^{2}}{4\cdot(x-x_p)} = p $ I'd like now to convert this expression to a polar representation. For this I got back to the basic rules: 1) $x = r\...
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2answers
25 views

Polar Equation to Rectangular

$$r=\frac{9}{4 \cos θ − 3 \sin θ}$$ How do I do this? (Equation is in polar form.) I have already tried to do this, but I don't know how to finish it.
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31 views

Help needed on evaluating the following double integral

Use polar coordinates to evaluate $\int\int_{D}\ x\ dA$ where $D$ is the region inside the circle $x^2 + (y-1)^2 = 1$ but outside the circle $x^2 + y^2 =1$. (It's like a crescent moon facing the ...
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1answer
28 views

how to write floor function vectors in polar coordinates

let $$\lfloor{x}\rfloor=y$$ And $$z=x-\lfloor{x}\rfloor$$ Plot the following vector in polar coordinates: $$x\hat{\imath}+(y/z)\hat{\jmath}$$ I know that while transforming from cartesian to polar we ...
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27 views

Find the center of mass of homogeneous object

I am asked to find the center of mass of this homogeneous object: Let's say that it's density is $k$ so the mass is $$ m = \int_{0}^{\pi} \int_{a}^{2a} k r drd\theta = \frac{3\pi a^2 k}{2} $$ So ...
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1answer
22 views

Find the area of this figure using polar coordinates (possible textbook mistake)

I am required to find the area of this region using polar coordinates: My setup is $$ A = \frac{1}{2} \int_{0}^{\phi} \left[ R \sin(\theta) \right]^2 d\theta = \frac{R^2}{4} \int_{0}^{\phi} \left[ ...
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1answer
20 views

How to show that a function is in a Sobolev space

This question is about the solution of exercise 1.20 in Elman, Silvester, Wathen. Finite Elements and Fast Iterative Solvers. (The first Chapter of the book is open access and available, for example, ...
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1answer
35 views

Find a polar equation for the curve of the given Cartesian equations: $y=x$, $4y^2 = x$ and $xy=4$

I am asked to find a polar equation for the curve of the given Cartesian equations: $y=x$, $4y^2 = x$ and $xy=4$. What I got here so far is $$ y = x\\ r \sin(\theta) = r \cos(\theta)\\ \boxed{\tan(\...
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3answers
57 views

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
32 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
50 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
16 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
21 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
53 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 d\theta$...
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0answers
36 views

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}} \int_{...
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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
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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2answers
78 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
63 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 just ...
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1answer
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
21 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
115 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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1answer
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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0answers
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
56 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: $$x=r*cosφ$$ $$y=r*sinφ$$ $$d(r*...
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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
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
65 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 u}{\...
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1answer
62 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
29 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 u}{\...
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0answers
39 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
32 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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0answers
44 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 ...