# 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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### Ellipse in polar coordinates

I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format - Where $2a$ is the total height of the ellipse and $2b$ ...
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### How to show the normal density integrates to 1?

How could you show that the normal density integrates to 1? $$\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi \sigma^2}} e^{-(x+\mu)^2 / \sigma^2} dx = 1$$
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I recently began experimenting with gnuplot and I quickly made an interesting discovery. I plotted all of the prime numbers beneath 1 million in polar coordinates such that for every prime $p$, $(r,\... 3answers 255 views ### Rigorous proof that$dx dy=r\ dr\ d\theta$I get the graphic explanation, i.e. that the area$dA$of the sector's increment can be looked upon as a polar "rectangle" as$dr$and$d\theta$are infinitesimal, but how do you prove this rigorously?... 1answer 2k views ### Heat equation in polar co-ordinates I was studying the heat equation, when i saw a new variant of it. Here's the statement: "the edge$r=a$of a circular plate is kept at temperature$f(\theta)$. The plate is insulted so that there is ... 2answers 103 views ### From Gravity Equation-of-Motion to General Solution in Polar Coordinates I'm having trouble getting the general solution of this differential equation. The gravitational equation of motion is, for constants$M$and$G$and position vector$\vec{r}, $$\frac{d^2}{d t^2}\... 1answer 99 views ### Integral formula for polar coordinates The polar coordinates of point x \in \mathbb{R} \setminus \{0\} are pairs (r,\gamma), where 0 < r < \infty and \gamma \in S^{d-1} = \{x \in \mathbb{R}^{d}\mid |x| = 1\}. These are ... 3answers 4k views ### Calculating a limit in two variables by going to polar coordinates. I have this limit to calculate:$$l=\lim_{(x,y)\to(0,0)}\frac{\sin(x^2y+x^2y^3)}{x^2+y^2}$$I solve it by going to the polar coordinates. Since (x,y)\to 0 means the same as \sqrt{x^2+y^2}\to 0, ... 1answer 529 views ### Partial derivatives and orthogonality with polar-coordinates We are stuck with this question here because I cannot understand the following results. I find it hard to visualize this, let alone deduce from that. How to do it? Objective to Attack The closely ... 1answer 152 views ### Really Stuck on Partial derivatives question Ok so im really stuck on a question. It goes: Consider$$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 \... 3answers 125 views ### 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 ... 2answers 18k views ### Why is dy dx = r dr d \theta [duplicate] Possible Duplicate: Explain \iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr I'm reading the proof of Gaussian integration. When we change to polar coordinates, why do we get ... 4answers 985 views ### How can the trefoil knot be expressed in polar coordinates? From Wikipedia, the parametric equations for a trefoil knot are \begin{align*} x(t) &= \sin t + 2\sin 2t \\ y(t) &= \cos t - 2\cos 2t \\ z(t) &= -\sin 3t. \end{align*} I am only ... 1answer 1k views ### Computing gradient in cylindrical polar coordinates using metric? I am trying to understand coordinate transformations properly (having studied some general relativity in the past). Let us consider the transformation from cartesian to cylindrical coordinates, x=\... 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 ... 2answers 108 views ### 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 ... 1answer 207 views ### Keeping the arc length constant between points in a spiral I'm making a visualization of points in a logarithmic spiral and want to keep the arc length between points (image particles) constant. I read that in an Archemedian spiral arc length is ... 1answer 76 views ### 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(\... 2answers 1k views ### Converting polar equation to cartesian coordinate polar equation and back again? OK, so I have the following polar equation:r = Θ/20$And I would like to translate this a little to the right, and down from the polar origin. Now, I figure since I know cartesian coordinate ... 2answers 757 views ### Find Cartesian equation of$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? $$r=\theta$$ ... 2answers 1k views ### How know which direction a particle is moving on a polar curve I have being doing problems from the released AP BC Calcululs Free-Response questions, and I have come to realize that I don't have a very good idea of explain or a deep understanding of how to tell ... 2answers 185 views ### Show that the parameterized curve is a periodic solution to the system of nonlinear equations First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the ... 2answers 47 views ### Explain finding the area of a region? How is the area of the region inside the lemniscate$r^2 = 6\cos(2\theta)$and outside the circle$r = \sqrt3$equal to$(3(\sqrt3) - \pi)$? Thank you for anyone that helps. 2answers 2k views ### Finding a point on Archimedean Spiral by its path length I've been curious about Archimedean Spirals and their relations to Sacks Spirals and prime numbers. I would like to draw some visualizations of the points with a given distance from the center, ... 0answers 42 views ### Does a plane curve with polar equation$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$have a name? Does a plane curve with polar equation $$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$$ where both$\lambda_i>0$have a name? It's very similar to hippopede, also known as lemniscate of Booth, ... 1answer 2k views ### Area Bounded by Polar Curves I am answering sample exams for my Calculus class and my attention was caught by the following item. Set-up the definite integral or sum of definite integrals equal to the area of the region above ... 5answers 212 views ### Points on$(x^2 + y^2)^2 = 2x^2 - 2y^2$with slope of$1$Let the curve in the plane defined by the equation:$(x^2 + y^2)^2 = 2x^2 - 2y^2$How can i graph the curve in the plane and determine the points of the curve where$\frac{dy}{dx} = 1$. My work: ... 2answers 69 views ### Given integral$\iint_D (e^{x^2 + y^2}) \,dx \,dy$in the domain$D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$Move to polar coordinates. Given integral$\iint_D (e^{x^2 + y^2}) \,dx \,dy$in the domain$D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$Move to polar coordinates. First of all I tried to find the domain of$x$and$y$: ... 6answers 2k views ### Simple partial differentiation$x = r\cos\theta$and$y = r\sin\thetaIf \begin{align} x &= r\cos\theta,\\ y &= r\sin\theta, \end{align} find $$\dfrac{\partial^2\theta}{\partial{x}\partial{y}}.$$ How can I find this partial derivative? I need to prove ... 1answer 3k views ### Horizontal and vertical asymptotes of polar curver = \theta/(\pi - \theta) \, , \, \in[0,\pi]$I as supposed to find the vertical and horizontal asymptotes to the polar curve $$r = \frac{\theta}{\pi - \theta} \quad \theta \in [0,\pi]$$ The usual method here is to multiply by$\cos$and$\...
I need to show that the del operator in 2D polar coordinates is $\nabla=e_r\partial_r+\frac{1}{r}e_r+\frac{1}{r}e_{\phi}\partial_{\phi}$. I try the following approach: \$\nabla=\partial_xe_x+\...