# Tagged Questions

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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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)} = \... 1answer 42 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, I ... 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: $$D= \{\space (x,y) \in \mathbb R ^2 \space |\space {1 \over 2} \le x \space,\space x \le y \le \sqrt 3 x\space,\space x^2+y^2 \le 1\space \}... 1answer 51 views ### When is {{x^2y} \over {(x^2+y^2)^\alpha}} continuous, using polar-coordinates Given$$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}$$For what values of \alpha, f is continuous in {(0,0)}? I set ... 1answer 116 views ### Pullback metric, coordinate vector fields.. I'm doing this computation on \mathbb{R}^3 with cylindrical coordinates (r, \theta, z), (which aren't defined on the whole of \mathbb{R}^3, but I don't care about that) and I seem to get a ... 0answers 20 views ### Find polar coordinates parameters given a cartesian point I'm stuck with a rather simple problem: I have the following polar coordinates equation where I know the values for the x and y terms and I want to find a, b and \Phi.$$x = a\cos(\Phi)... 0answers 29 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: $$\... 2answers 72 views ### Expressing (-8)^{\frac13} in polar form I want to express (-8)^{\frac{1}{3}} in polar and cartesian coordinates. What I did was to solve the equation -8 = r^3e^{3i\theta}= r^3(\cos(3\theta)+i\sin(3\theta)) which implies that I must ... 2answers 13 views ### Expressing a value in polar and cartesian coordinates I have to express the value of \sqrt{i} in polar and cartesian form. I really don't know where to start this problem, any hint could help me please! 0answers 26 views ### Relationship between Normal coordinates and Spherical Coordinates I am using the following coordinates on S^3: (\psi, \theta, \phi) where$$\begin{cases}x_0 = \sin\psi,\\ x_1 = \sin\psi \cos\theta,\\ x_2 = \sin\psi \sin\theta \cos\phi,\\ x_3 = \sin\psi \sin\... 3answers 29 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$y = \sqrt{b}\sin t$... 0answers 26 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 ... 4answers 69 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 ... 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 $$\begin{split} dx &= \cos\theta \, dr - r\sin\theta \, ... 2answers 103 views ### How would one convert the cartesian expression y=1/x to polar form? How would one convert the cartesian expression y=1/x to polar form? I'd really appreciate a step-by-step solution so I can apply the same principle to other problems. Thanks! 2answers 26 views ### Equations of motion into Binet's equation I have been given these two equations in polar coordinates: m(r''− rθ'^2) =−f, m(r''θ + 2r'θ') = 0 And have been told that I need to differentiate to show the angular momentum L=mr^2θ' is ... 2answers 52 views ### How to reverse the integration order of the double integral \int_{\theta=0}^{2\pi}\int_{r=0}^{1+\cos\theta}r^2(\sin\theta+\cos\theta)drd\theta. I am given the integral$$ \iint\limits_H \, (x+y) \mathrm{d} A $$where H is the area of the cardioid r=cos(\theta)+1. I have translated the double integral to polar coordinates in order to solve ... 1answer 21 views ### Convert \int_0^1dx \int_0^{x^2}f(x,y) dy to polar integration Converting \int_0^1dx \int_0^{x^2}f(x,y) dy to polar coordinates: (r \cos\theta)^2 = r \sin\theta, so r= tg\theta\sec\theta, then the result is,$$\int_0^{\frac\pi4} d\theta \int_0^{tg\theta\... 2answers 311 views ### Arc length of Archimedes Spiral$ r = \theta $from$ 0 \le \theta \le 2\pi$The equation of the Archimedes spiral is given by $$r = \theta$$ The formula for calculating the Arc Length is given by $$L = \int^b_a\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta$$ The ... 0answers 78 views ### Polar System with Short Answers, How$U(0, \theta)=\pi$will be calculated? I read some notes on Laplace. I ran into a short answer question as follows. We have a Laplace equation in Polar Systems:$\frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial u}{\partial r})+\...
Find the area of the region bounded by: $$r=5\cos(10\theta),~~~~~ 0 \leq \theta \leq 2\pi$$ When I did this, I got $\frac{1}{2\sin(20\pi)}-\frac{1}{2\sin(0)}$ getting $0$, is this correct?