# 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 ($\theta$).

2,611 questions
Filter by
Sorted by
Tagged with
17 views

33 views

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 ...
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: $\space$...
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 ...
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 ...
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-...
40 views

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 $t = 2\phi$ ...
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 ...
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 $t$,...
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. ...
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 $$r=\sqrt{9+4(\cos(t)+\sin(t)}$$ ...
21 views

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