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 ($\theta$).

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What is $b$ in $r=b\theta$ of Archimedean spiral?

This Wikipedia entry says Equivalently, in polar coordinates $(r, θ)$ it (Archimedean spiral) can be described by the equation $r = b\theta$, with real number $b$. Changing the parameter $b$ controls ...
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Let $P$ be a point in the first quadrant that lies on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$.

Let $a$ and $b$ be positive numbers such that $a>1$ and $b<a$. Let $P$ be a point in the first quadrant that lies on the hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$. Suppose the tangent to ...
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Continuity Of Argument Function.

Fix $m\in \mathbb R$. Define $f_m :\mathbb R^2 \setminus\{(0,0)\}\rightarrow(m,m+2\pi]$ $~~$as $(x,y) \mapsto$ argument of $(x,y)$ in $(m,m+2\pi]$. i.e $$(x,y)=\left(\cos(f_m (x,y)),\sin(f_m (x,y))\...
Meet Patel's user avatar
2 votes
1 answer
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orienting a point in polar coordinates along a particular unit vector

I have the center of a circle $\vec{c}$ in 3 space and the radius $r$. I also have a unit vector $\hat{v}$ defining the orientation of the plane of the circle. I wish to parameterize this circle and ...
Stan Shunpike's user avatar
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50 views

Integrate $||x||^{-\alpha}$ on $[a,b]\times [c,d]$

Consider $\alpha >1$ and let $a,c>0$ and $a<b$, $c<d$. How can I compute $$\int_{[a,b]\times [c,d]} ||(x_1,x_2)||^{-\alpha} dx=\int_{[a,b]\times [c,d]}\frac{1}{(x_1^2+x_2^2)^{\frac{\alpha}{...
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Circle rolling between two functions

Consider a circle of radius $r_s$ that is tangent to two curves $r(\theta)$ and $R(\theta)$ at points $E_1, E_2$ respectively, defined in polar coordinates. Knowing the function $r(\theta)$, find the ...
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Find the area between $r = \tan ( \theta )$ and $r = -\theta$ for $0 \leq \theta \leq 2 \pi$. Round your answer to the nearest thousandth

I was trying to solve this problem, but I still can't wrap this around my head. When plugging this into a graphing calculator, I found three intersections, and since I couldn't find them algebraically ...
zm azad's user avatar
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Rotating the axes and changing variables

Without using matrix inversion, what is the easiest way to show that, for variables $u$ and $v$ corresponding to rotating the $x-y$ axes by $\theta$, we have $$ x=u\cos \theta -v \sin \theta\\ y=u\sin\...
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Centroid of a parabolic arc

Find the centroid $C=(\bar{x},\bar{y})$ of the parabolic arc $y=16-x^2$ over $[-4,4]$. From symmetry, $$\bar{x}=0$$ To find $\bar{y}$, substitute $\tilde{y}=y$, $dL=\sqrt{1+4x^2} dx$ in $$\frac{\...
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Constructing Logarithmic Spirals with Rays

This page states "The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray... As the number of ...
origamifreak2's user avatar
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Area of a circle described in polar coordinates using a Double Integral

I want to find the area of circle specified as a function of angle $\theta$, where the circle can be described as $r(\theta)=a$ means for every angle $0\le\theta\le2\pi$ the distance from the origin ...
Jeffy James's user avatar
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Angle between two rays given, to find a curve with constant sum of intersection angles

Two given lines $AB, AC$ including $\angle A= 60^{\circ}$ between them are intersected by a variable smooth continuous curve (red) so that angle sum $ \beta+\gamma = 150^{\circ} $ is always constant....
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Polar coordinates: how to plot straight line at $n$ degrees?

What equation represents a straight line at $n$ degrees/radians in polar coordinate system? I know in Cartesian coordinates: $y = \sin\frac{2\pi}{3}x +0$ Polar conversion: $x = r \cos\rho$ and $y = r \...
Nick's user avatar
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Find $x$ such that $\cosh(a + bx) + 1 = cx$

I need to find an analytical solution for $x$ to: $$ \cosh(a + bx) + 1 = cx $$ where a,b and c are real parameters. I have tried to tackle this geometrically, by splitting the problem into finding ...
Gabriele Vecchio's user avatar
2 votes
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Average over sphere identity

In Lieb-Loss Analysis Theorem 9.7, they write in the proof, that for $\mathbb{R}^3$ the following identity holds $$\frac{1}{4\pi} \int_{\mathbb{S}^2}d\omega \frac{1}{|r\omega-y|} = \min\big(\frac{1}{...
MathematicalMoose's user avatar
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1 answer
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Show that the tangent to $\left(0,\frac{\pi}{2}\right)$ on the polar curve with equation $r^2=a^2\sin 2\theta$ is perpendicular to the initial line

Find the polar coordinates of the points on $r^2 = a^2 \sin 2\theta$ where the tangent is perpendicular to the initial line ($\theta=0).$ I guess they are assuming $a>0.$ The answer in the back of ...
Adam Rubinson's user avatar
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Connection between two derivations of velocity in polar coordinates

I'm having some trouble connecting two different derivations of velocity in polar coordinates: a geometric method and one involving the product rule of derivatives. Both arrive to the same result, ...
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How to realize the function $\bar f(x)=\frac{1}{2\pi}\int_0^{2\pi} f(x,\theta)\;d\theta$?

Maybe this sounds quite silly and maybe I should take a break from math during Christmas and I apologize in advance, but how should I realize this function: $$\bar f(x)=\frac{1}{2\pi}\int_0^{2\pi} f(x,...
kaithkolesidou's user avatar
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Polar coordinates limits multivariable calculus [duplicate]

I’m studying multivariable calculus. When trying to solve a limit $\lim_{(x,y)\to(x_0, y_0)}f(x,y)$ with $f:\mathbb{R}^2\to\mathbb{R}$, our professor told us that, in some cases, the limit is more ...
selenio34's user avatar
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calculating the Jacobian in polar coordinates geometrically

The task sounds like this: “geometrically calculate the Jacobian of polar substitution.” In this case, we get the expression dxdy=rdrd(theta)+ 4s, where 4s are the remaining pieces when inserted. But ...
slaid world's user avatar
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Calculate the area bounded by two curves via double integration

I need to find the area of the middle part bounded (or between) 2 curves: $ x²+y²=1$ and $ 4x²-y²=1$. I have the graphic of the middle part (the part, which I need to calculate the area for it), but I ...
Mohaboko's user avatar
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On the gradient and Laplace operator of the cylinder

This is about something that I found intriguing while computing some geometric identities for the cylinder of radius $r$ which we denote by $C = \{(r\cos(\theta),r\sin(\theta),z) : \theta \in [0,2\pi),...
user57's user avatar
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Double integral of $\frac{x}{1+x^2+y^2}$

I'm trying to crack an integral problem whose answer has been lost: $$ I:=\int_0^\frac{1}{\sqrt{2}}\int_\sqrt{1-x^2}^\sqrt{3-x^2}\frac{x}{1+x^2+y^2}dydx+\int_\frac{1}{\sqrt{2}}^\sqrt{\frac{3}{2}}\...
Boar's user avatar
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Proof of Area Under Polar Curves

The books use the area of sectors to derive the formula for the area under the polar curves. I couldn't understand how when the number of sectors reach infinity, the area of the region created by each ...
WBI's user avatar
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Double Integration - Sum approximation - Peter Saulson article on terrestrial gravitational wave antenna

I'm currently reading an article on terrestrial gravitational noise done by Peter Saulson. He approximates a sum (Eq.4) by an integral with a inner cutoff radius of lambda/4 (Eq.5). I don't manage to ...
QuentinCP's user avatar
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Is there a general solution for the integral that gives the force of gravity created by a 2-Dimensional body defined by a sum of cosines?

I'm trying to solve the integral which would provide the force of gravity in a 2-Dimensional universe on an object being influenced by a body with uniform density who's perimeter is defined by ...
ImIsaacKane's user avatar
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Whats the difference between using line integral of the first kind and normal integral to find the mass of the given curve

I was working on a question that asked me to find the mass of a given curve using a specified density function. My initial approach was to use the line integral of the first kind, which intuitively ...
Raymonk Surya's user avatar
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general parametric form of curvature in polar coordinate

I saw this in another question $$\kappa(\theta) =\frac{x'(\theta)y''(\theta)-x''(\theta)y'(\theta)}{(x'(\theta)^2+y'(\theta)^2)^{3/2}}$$ Polar curvature I know that the general form of parametric ...
Ao Bo's user avatar
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3 votes
3 answers
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Father duck swim along $y=-x^2$,baby duck swim along $y=x^2$, Father duck’s movements parameterized by: $𝒓(𝑡)=(𝑡−5,−(𝑡−5)^2)$, equation for baby?

A father duck and a baby duck are swimming along opposing parabolic paths, with the father along $𝑦 = −𝑥^2$ and the baby along 𝑦 = $𝑥^2$. The father duck’s movements can be parameterized by: $𝒓_𝟏...
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About the Graph of theta=const in the Polar Coordinate System

Sorry for this question but I want to be get clarified. Lets look at the image below. My question is this. Why not include the point of origin when in fact the polar coordinates $(0,1\ \text{rad})$ ...
Juniven Acapulco's user avatar
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Compare function in cartesion coordinate system and in polar coordinate system

I plotted the function $$f(x, y) = 0.5*x^2+y^2$$ in Geogebra. Also i plotted this function but in polar coordinate system $Surface(u*\cos(v),0.5*u*\sin(v),u^2,v,0,2*π,u,0,3)$. Unfortunately, the plot ...
Daniil Stepanov's user avatar
3 votes
0 answers
58 views

Integration over ball

I have the following exercise: Evaluate the multivariate integral $$\int_Af(\boldsymbol x)\,\mathrm d\boldsymbol x$$ numerically, where $f:\mathbb R^p\to\mathbb R$ is a known function and $A := \{\...
Syd Amerikaner's user avatar
4 votes
1 answer
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Is knowing unit basis vector enough to specify a coordinate system?

In orthonormal curvilinear coordinate system, we define unit basis vector as $$\mathbf{\hat{e}}_u = \frac{1}{h_u} \frac{d\mathbf{r}}{du},$$ where $h_u = |\frac{d\mathbf{r}}{du}|$. Suppose we only know ...
Sean's user avatar
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2 votes
1 answer
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Area Calculation of Region A (Using Double Integral)

I'm calculating the area of the region $A$ defined by the following constraints: $$ A = \{(x, y) \mid x^2 + y^2 \geq 2, \; x^2 + y^2 \leq 2x, \; y \geq 0\} $$ To calculate the area of the region $A$, ...
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Two ways of finding families of curve orthogonal to circle

Given the family of curves $x=r\cos(\theta)$ and $y=r\sin(\theta)$ with $\theta \in [0,2\pi)$, I wish to find another family of curves orthogonal to it. One method is as follows: $$x^2+y^2=r^2 \...
Sean's user avatar
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Iterated integral change the order and go to polar coordinates

I have an iterated integral with these two homework assignments on it : (1) change the order of integration (2) go to polar coordinates and set the limits of integration according to new ones ...
Nick Schemov's user avatar
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2 answers
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Area between a circle and an spiral.

I have to evaluate the following area: Since the area of the circle is $16\pi$, I thought about evaluate the area of the spiral from $0$ to $4$: $$\frac{1}{2}\int_{0}^{4}\theta^2 d\theta=\frac{32}{3}$...
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Solve parameterized line integral

In my physics problem I've modelled the following integral. \begin{equation} p(X,Y,x_0,y_0) = \int_{0}^{2\pi} \frac{1}{\sqrt{(X-R\cos(\Psi))^2+(Y-R\sin(\Psi))^2}} \delta(g) d\Psi \end{equation} Where $...
Matthew James's user avatar
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Dot Product with Partial Differential Question

Given: $$\vec{A}(r,\theta)$$ $$\vec{B}(r,\theta)$$ Is it always true that: $$ \left(\vec{A}\frac{\partial}{\partial{\theta}}\right)\bullet\vec{B}\overset ? =\bigg(\vec{A}\bullet\frac{\partial\vec{B}}{\...
John's user avatar
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1 vote
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Constructing $f : \mathbb{R}^2 \setminus \{ (x,0) \mid x \leq 0 \} \to (-\pi, \pi)$ using polar coordinates

I'm trying to construct a well-defined $C^\infty$ function $f(x,y) : \mathbb{R}^2 \setminus \{ (x,0) \mid x \leq 0 \} \to (-\pi, \pi)$ via polar coordinates, which cannot be continuously extended to $\...
GreenCoffee248's user avatar
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Reference request for texts on polar coordinates.

I am looking for books, papers, and other texts on polar coordinates. I want such a text to rigorously define polar coordinates, and also derive the polar coordinate equations for common geometric ...
user107952's user avatar
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Laplace equation in polar coordinates Dirichlet Problem

I am not sure how to prove the following. (P591 Problem 4(b),Problem Set 12.10 "Advanced Engineering Mathmatics" by Erwin Kreyszig 10th ed.) Assuming that termwise differentiation is ...
KenN's user avatar
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3 votes
3 answers
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Show that $y=\frac{1}{x}$ is a hyperbola

I am trying to write $y=\frac{1}{x}$ in the standard form for a hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a>0$, $b>0$) but I can't figure out the last step. My book says to First, rotate ...
msb15's user avatar
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What would be the new Equation of motion if the magnetic field's origin is shifted from the origin of a co-rotating spherical polar coordinates?

The equation of motions due to the dipole magnetic force of a planet in a frame corotating with the planet and origin at the centre of planet assumed to be sphere components wise are given as below: \...
Lunthang Peter's user avatar
1 vote
3 answers
58 views

How can an angle have a direction in polar systems?

I was learning about torque, angular momentum etc. using polar coordinates. In the lecture, I learned that the unit vectors are $\hat{𝜑}$ and $\hat{r}$, where 𝜑 is the angle and r is the distance ...
Bruce M's user avatar
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1 answer
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Converting cartesian to polar form

Given $$y^2 = x^2 - x^4,$$ how do you represent this in polar form? I tried substituting $$x=r\cos \theta$$ and $$ y= r\cos \theta$$ which gave me $$r = \sqrt{\frac {\cos^2 \theta - \sin^2 \theta} {\...
William's user avatar
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How to get to $dr=\frac{x}{r}dx+\frac{y}{r}dy$ and $d\theta=\frac{-y}{r^2}dx+\frac{x}{r^2}dy$

I'm reading a short text called "The Planimeter as an Example of Green's Theorem", written by Ronald W. Gatterdam. In the text, there is a figure of an idealized planimeter, shown below. ...
Breno's user avatar
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Evaluation of the integral. (Using Polar coodinates.)

Can anyone prove this equality? $$\int_{\mathbb R^2}f(x)\int_{|y|<1}\frac{e^{ix\cdot y}\ -1}{|y|^2}dydx=\int_{\mathbb R^2}f(x)\int_0^1 \frac{1}{r}\int_0^{2\pi}(e^{i|x|r\cos\theta}\ -1)d\theta dr dx,...
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Why are my solutions to the Christoffel symbols of polar coordinates recipricoled?

I just gave myself some practice with Christoffel symbols, using the polar coordinates formula of $[x, y] = [r\cos\theta,\,r\sin\theta]$. I then calculated the Christoffel symbols, and I got six ...
Alexandra's user avatar
  • 325
2 votes
1 answer
77 views

Conservation of swept area (Kepler's $2$nd law), rigorous proof

Briefly, in class we proved Kepler's $2$nd law like this: We've some random trajectory and two position vectors, $\mathbf r$ and $\mathbf{r}+d\mathbf{r}$. Supposing $dr$ is small, we can approximate ...
Joan S. Guillamet F.'s user avatar

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