Questions tagged [polar-coordinates]

Question 1

What is the need for different sets of unit vector for different locations (positions) in Polar Coordinate System ? Just as the two fixed unit vectors in Cartesian Coordinate System, why cannot we ...

Question 2

I was playing around in Desmos looking at rose-shaped curves, a family of curves with polar equation $$ r = \cos n \theta, \ \ \ \ \ n \in \mathbb{N}. $$ The number of petals on this rose-curve is ...

Question 3

This is my first proof and is most likely going to be crude. Please while reading comment tips about ways to improve my writing. Polar coordinates are transformed through: $$x=r\cos(\theta) \qquad y=r\...

Question 4

From this answer I have that $ \int_Yf(y)\,\mathrm{d}(g\mu)(y)=\int_Xf(g(x))\,\mathrm{d}\mu(x)$, where $g$ is a map between measurable spaces and $g\mu$ is the image measure. With $X=[0,r]\times[0,2\...

Question 5

Problem: For a function $f(x,y,z)$ and a rotational change of coordinates $(x,y,z)\to (u,v,w)$, the following relation holds $$\frac{\partial^2 f}{\partial x^2}+\frac{\partial^2 f}{\partial y^2}+\frac{...

Question 6

I'm working with a scriptable 3-D rendering tool that, due to various rounding and binary representation errors in point arithmetic will throw errors at extremely rare but always inopportune times. ...

Question 7

Given a circle with radius $r$ centered on the origin at (0,0) and, in polar coordinates: a start point, at radial distance $p$ (with $ 0 < p < r$) from the origin, and angle $\alpha$ an end ...

Question 8

Supposing to have the complex number $$z=-4 - 8 i$$ Having \begin{equation} \varrho = |z| = \sqrt{(-4)^2 + (-8)^2} = \sqrt{16 + 64} = \sqrt{80} = 4 \sqrt{5} \end{equation} \begin{equation} \arctan\...

Question 9

Find the $n^{th}$ Derivative of $$f(x)=\tan^{-1}\left(\frac{2x}{1-x^{2}}\right)$$ in terms of polar coordinates $(r,\theta)$ of $x=re^{i\theta}$. My Approach $$f(x)=\tan^{-1}\left(\frac{2x}{1-x^{2}}\...

Question 10

I have a complex number in cartesian form that I am trying to express in polar form. I haven't seen a form like this before but I assumed I would be able to solve it by multipyling the denominator and ...

Question 11

Let $K\subset \mathbb{RP}^2$ be closed, convex set that does not contain a whole line, and is not all of $\mathbb{RP}^2$ with nonempty interior. Let $\gamma=\partial K$ be a $C^3$ curve. Then for any ...

Question 12

I need to find the asymptote of $r \log_e(\theta)=a$ I understand what is going on. As $\theta$ goes from $0^+$ to $1$, $r$ goes from $0^-$ to $-\infty$. I tried to write the equation as a function of ...

Question 13

Compute the limit (if exists) $$\lim_{(x,y)\rightarrow (0,0)}e^{\frac{-y^2}{x^4}}\sqrt[3]{y}.$$ My attempt: We can pass through polar coordinates: $$\Bigg\lvert e^{\frac{-\sin^{2}(\theta )}{\rho^{2}\...

Question 14

The slope of a polar function $f$ is generally given by the formula $f'(x) = \frac{f'(\theta)\sin(\theta) + f(\theta)\cos(\theta)}{-f(\theta)\sin(\theta) + f'(\theta)\cos(\theta)}$ However, the rate ...

Question 15

In the manifold $$ M=\{x\in\mathbb{R}^2:\ |x|>a\}\quad(a>0), $$ you can write $M\cong (a,\infty)\times S^1$ with polar coordinates $(s,\theta)$ and take a conformal rescaling of the Euclidean ...

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Questions tagged [polar-coordinates]

Unit Vectors in Polar Coordinate System [closed]

Why does every rose curve contain a regular polygon?

The Jacobian and Integration of $n$-dimensional polar coordinates

How do I prove the change of variables into polar coordinates using measure theory?

If an operator is invariant with respect to 2D rotation, is it also invariant with respect to 3D rotation?

How to overcome Cartesian/Polar round trip anomalies in 64-bit IEEE floating-point numbers

Drawing a smooth curve with the least maximum curvature from a point within a circle to its boundary

Use $\operatorname{Arg}(z)$, $\vartheta$ or $\arg(z)$ for the angle $\varphi$ in the polar form $w = r e^{i \varphi}$

Find the $n^{th}$ Derivative of $\tan^{-1}\left(\frac{2x}{1-x^{2}}\right)$ in terms of polar coordinates of $x$ [closed]

Expressing $\frac{\sqrt{2} + j\sqrt{2}}{1 + j\sqrt{3}}$ in its polar form

Does the butterfly property characterize conics (including degenerate)?

Asymptote of $r \log_e(\theta)=a$

Limit of 2 variable function with exponential function

Radially outward tangent of polar curve

In $\Bbb{R}^2\setminus\Bbb{D}^2$ with warped metric write the limit of θ-coordinate as the geodesic reaches the boundary in terms of initial data

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