Articles of polar coordinates

How have I incorrectly calculated the area A inside the curve $r =1$ and outside the curve $r = 2\cos(\theta)$

I hope this isn’t against guidelines but I want to show how I solved this on white paper, it might be clear where my logic went wrong in solving. Find the are A of the region inside the curve $r = 1$ and outside the curve $r = 2cos(\theta)$ Thank you

About polar coordinates in high dimensions

I’m trying to understand a proof in Michel Willem, Functional Analysis — Fundamentals and Applications, Birkhäuser. The book defines: $$\int_{\Bbb S^{N-1}}f(\sigma)\,d\sigma=N\int_{B_N}f\left(\frac{x}{|x|}\right)dx.$$ And then goes on to proving: $\textbf{Lemma 2.4.7.}$ Let $u\in{\cal K}(\Bbb R^N)$. Then (a) for every $r>0$, the function $\sigma\mapsto u(r\sigma)$ belongs to $C(\Bbb S^{N-1})$; (b) $\displaystyle\frac{d}{dr}\int_{|x|<r}u(x)\, dx=r^{N-1}\int_{\Bbb S^{N-1}} u(r\sigma)\,d\sigma;$ (c) $\displaystyle \int_{\Bbb R^N}u(x)\, […]

Find Cartesian equation of $r=\theta$

I solved this problem, but I’m not sure my answer is correct as it seems very complex (compared to the polar equation). Did I make some mistake along the way or is it the right solution? $$r=\theta$$ $$\sqrt{x^2+y^2}=\arctan\bigg(\frac{y}{x}\bigg) \tag{$r^2=x^2+y^2 \to r=\sqrt{x^2+y^2}$}$$ $$x^2+y^2 = \bigg(\arctan\bigg(\frac{y}{x}\bigg)\bigg)^2$$

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 contradiction. The problem is as follows. Make the following change of coordinates: $$\xi = r,~~\eta = \theta – z,~~\zeta = z.~~~(1)$$ My problem […]

Area and Polar Coordinates

Would anyone be able to help me with this problem? I think I know the area formula in polar coordinates that should be used: the antiderivative of ((1/2)r^2 dtheta) from alpha to beta but I’m not really sure how to get the area of the removed part. Thank you for any help that could be […]

Determining the parameters for a spiral tangent to an arc and intersecting a specified point in 2D.

I am attempting to determine the polar equation for a constant pitch spiral of the that starts at an arbitrary point $(r_1,θ_1)$ and is tangent to an arc of radius $a$ centered at $(r_c,θ_c)$ at point $(r_2,θ_2)$. The spiral I need to generate needs to have constant pitch and be centered at $(0,0)$ but is […]

A couple of GRE questions

Look for help with the following GRE questions Question 1. If $C$ is the circle in the complex plane whose equation $|z|=\pi$, oriented counterclockwise, find the value of the integral $\oint_C(\cos z-z\cos\frac{1}{z})dz$. Question 2. How to show the sequence $\{x_n\}_{n=1}^\infty$ definted by $x_{n+1}=\frac{1}{2}(x_n+\frac{2}{x_n})$, $x_1\ne 0$ converge. Question 3. Let $L$ be the curve whose equation […]

Partial derivatives and orthogonality with polar-coordinates

We are stuck with this question here because I cannot understand the following results. I find it hard to visualize this, let alone deduce from that. How to do it? Objective to Attack The closely Related Problems with Orthogonal Basis and Dot Products In Polar-coordinates $\left(\hat{e}_{r}\partial_{r}\right) \cdot \left(\frac{1}{r}\hat{e}_{\theta}\partial_{\theta}\right)= 0$ $\left(\frac{1}{r}\hat{e}_{\theta}\partial_{\theta}\right) \cdot \left(\hat{e}_{r}\partial_{r}\right) = \frac{1}{r} \partial_r$ […]

Does the inverse function theorem fail for $\frac {\partial r}{\partial x}$

This is a follow-up to a question that I answered (though, not well enough). Why is it that $\frac {\partial r}{\partial x} = \cos(\theta) = \frac {\partial x}{\partial r} = \frac {\partial}{\partial r}(r\cos(\theta))$? $r$ and $x$ should be continuously differentiable so the inverse function theorem should apply. And I don’t think $r$ should be a […]

Really Stuck on Partial derivatives question

Ok so im really stuck on a question. It goes: Consider $$u(x,y) = xy \frac {x^2-y^2}{x^2+y^2} $$ for $(x,y)$ $ \neq $ $(0,0)$ and $u(0,0) = 0$. calculate $\frac{\partial u} {\partial x} (x,y)$ and $\frac{\partial u} {\partial y} (x,y)$ for all $ (x,y) \in \Bbb R^2. $ show that $ \frac {\partial^2 u} {\partial x […]