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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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Local polar coordinate system to global polar coordinate system

Let's have a point located in the local polar coordinate system at $(r,\phi)$. The origin of this local system is located at $(R_0,\Phi_0)$ in a global coordinate system. I would like to express $(r,\...
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62 views

Transform $dx/dt$ to $dr/dt$ polar coordinates

I've had to screenshot the question and post it as a photo
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1answer
56 views

Proving that $\lim_{(x,y)\to(0,0)} (x^2+y^2)^{x^2y^2}=1$ using polar coordinates

Am I doing this right? I rewrite the function as follows: $$(r^2\cos^2\theta+r^2\sin^2\theta)^{r^4\cos^2\theta\sin^2\theta} \stackrel{\text{various trig identities}}{=} r^{\frac{1}{4}r^4\sin^2 2\...
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1answer
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How to describe the graph of a spiral whos origin moves over time?

Problem: I am trying to figure out how I would graph the absolute coordinates of the spiral r = 14 + 0.69θ^1.8, θ = 4π to 0 when the relative system starts with its origin at absolute (0,0) at θ =...
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1answer
35 views

How can I transform $r=\theta$ to a parametric equation? [on hold]

How can I transform the polar equation $r=\theta$ into a parametric equation? Thank you.
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Polar Coordinates for Two Parallel Lines which has to follow an Arc

What I have What I want to obtain Hi, I have these two lines (What I have ), which one is described by two points with the next polar coordinates: LINE 1: Point 1: ( R1*cos(alfa1) ,R1*sin(alfa1) ,...
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1answer
33 views

Polar and parametric curves

I was solving a calculus problem on polar coordinates and I came across with some doubts, I don't know how to solve it. It says: "Given the curve $C: (x+1)^2+y^2=1$ parametrize the arc of a curve that ...
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Volume by rotation using polar coordinates

I have been trying to solve an exercise taken from the legendary brazilian book "Um Curso de Cálculo Volume 3" about change variable in double integral. This exercise literally says: Consider the ...
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1answer
30 views

Polar form of elliptic curve?

My instructor asked us to find the polar form of the elliptic curve defined by the equation $$y^2=x^3+ax+b$$ What I did: Using $x=r\cos\theta$ and $y=r\sin\theta$, I got $$r^2\sin^2\theta=r^3\cos^3\...
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1answer
35 views

How to find area between two functions of a polar curve?

Find the area lying outside $r=2\cos\ \theta$ and inside $r=1+\cos\ \theta$. Using the equation $\int_{\theta}^{2pi}(r^2/2) d\theta$, I would get $\int_{0}^{2\pi}((1+\cos\ \theta)^2/2)-((2\cos \ \...
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1answer
26 views

Polar co-ordinates, Jordan form, Axler textbook

I am trying to solve this Linear algebra question but I am unsure on how to proceed and get stuck. Define a three-dimensional ``Givens rotation'' in the 1-2 plane by $$M := \left( \begin{array}{ccc} ...
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Cross product in polar coordinates

$$\vec{r}\times m(\dot{r}\hat{r}+r\dot{\theta}\hat{\theta})=mr\hat{r}\times (\dot{r}\hat{r}+r\dot{\theta}\hat{\theta})=mr^2\dot{\theta}(\hat{r}\times\hat{\theta})$$ In the first equation I know we ...
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Push-Forwards and Pull-Backs: Polar to Rectangular Coordinates

Let $x=r*cos(\theta)$ and $y=r*sin(\theta)$ represent the polar coordinates function $\mathbf f(r,\theta):\mathbf R^2\rightarrow\mathbf R^2$. Compute $\mathbf f_*(r\frac{\partial}{\partial r})$ and $...
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2answers
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Converting a probability function from Polar coordinates to Cartesian coordinates

I am confused about converting a Probability Density Function from Polar coordinates to Cartesian coordinates. Here is an example: In Polar coordinates, we can have a Gaussian probability function: ...
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1answer
72 views

How to prove the integral converges?

Let $(\mathbf M'.\hat{\mathbf n})$ and $f(R,\theta)$ be a continuous function of $R$ and let $f(0,\theta)=0$. Then how shall we prove the following improper integral converges: $\displaystyle\lim \...
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1answer
44 views

Double integral from Cartesian to polar coordinates

I want to write the following double integral $$\int_0^2 \int_{0}^{\sqrt{1-(x-1)^2}} \frac{x+y}{x^2+y^2} \, \mathrm d x \mathrm d y$$ in polar coordinates. The region is a circle centered at $(1,0)$ ...
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2answers
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Converting double integral from cartesian to polar coordinates

I want to find $\displaystyle \int_1^4 \int_{0}^{\sqrt{x}}\exp(y/x)\,\mathrm dy\mathrm dx$ by transforming the integral to polar form. The region of integration is a part of the area under $\sqrt{x} $...
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Find the volume of solid region bounded by three cylinders.

The equations of the 3 cylinders are given by $x^2+y^2=1$, $y^2+z^2=1$ and $x^2+z^2=1$. While it is common to solve it in cylindrical coordinates via triple integral, I would like to know how to ...
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2answers
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Problem with intersection areas of polar curves

The original problem: Consider the curves $r = 3 \cos \theta$ and $r = 1 + \cos \theta$. (a) Sketch the curves on the same set of axes. (b) Find the area of the region inside the curve $r ...
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What is the “Full Cycle” of a Polar Curve?

In order to find the arc length or area etc of a polar curve, you must integrate from $\theta_1$ to $\theta_2$. However, I'm having trouble finding the values of $\theta_1$ and $\theta_2$. I know ...
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2answers
32 views

Finding Bounds for Area Between Polar Curves

I'm trying to find the area of the region both inside the circle $r= sinθ$ and outside the circle $r=√3 cosθ$ (both equations are in polar coordinates). Here is what it looks like: The two graphs ...
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Why are we not using Dirac delta and ignoring the contribution to the surface integral from the point $r=0$?

Let $V'$ be the volume of dipole distribution and $S'$ be the boundary. The potential of a dipole distribution at a point $P$ is: $$\psi=-k \int_{V'} \dfrac{\vec{\nabla'}.\vec{M'}}{r}dV' +k \oint_{...
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1answer
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Tangent at the pole for the equation $r = 2(1 - \sin\theta)$

I was asked to find the tangents at the pole for the following equation: $r=2(1-\sin\theta)$. I understand that the requirements for tangency at the pole are $f(\theta)=0$ and $f'(\theta) \neq 0$. I ...
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1answer
25 views

Tangents to Polar Curves

I am given the polar curve $r=3cos(θ)$. I am to list all of the points (three in total) $(r,θ)$ where the tangent line is horizontal. From $r=3cos(θ)$, I was able to derive that $$x = rcos(\theta) = ...
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How to convert from complex number to polar and vice versa? [duplicate]

My professor is constantly converting between complex numbers and polar coordinates throughout the specific lessons we're learning in my ECE courses. I'm not too sure on how she's doing it because I ...
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3answers
45 views

Finding the roots of a complex number [duplicate]

I was solving practice problems for my upcoming midterm and however I got stuck with this question type. It is asking me to find all roots and then sketch it. $(1+i\sqrt{3})^{1/2}$ How do we ...
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1answer
16 views

Conversion from Cartesian to polar coordinates

I have some difficulty with converting this implicit, Cartesian function into polar form: $\left(y^2+x^2\right)^2=2\left(x^2-y^2\right)$ I know that, in order to attempt to convert it, I need to use $...
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Prove that there exists a $m×m$ lattice square in the $x-y$ plane such that none of its coordinates are visible [duplicate]

Call a lattice point 'visible' if the $gcd$ of its coordinates is 1. Then there exists a $m×m$ square in the $x-y$ plane such that none of its coordinates are visible. You can actually define such ...
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When this polar curve intersects with the $y$-axis, why is the value of the angle $\frac{\pi}{2}$?

I'm currently learning area bounded by polar curve on Khan Academey, in one of the exercise, they asked I don't really understand how the value for beta is obtained (or to be more precise, the ...
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1answer
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Mean distance between two points in a square polar coordinates conversion

In this video, around the 5:20 mark, YouTuber MindYourDecisions converts the double integral from rectangular coordinates to polar coordinates so as to make the evaluation easier. However, I fail to ...
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1answer
35 views

What are the polar coordinates of $(2\sqrt3, 2)$?

My answer to this is $(4,\frac{π}6)$. But a calculator said that $(-4,\frac{7π}6)$ is also an answer, and there are infinitely many solutions. Is that correct?
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1answer
29 views

Mathematical proof regarding angle mirroring

I'm trying to find a proof for a statement that is made in Griffiths' Introduction to Electrodynamics. It can be stated as follows (in my own words): Let $P$ be a point such that its angle in polar ...
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Why does $\int_0^R 2 \pi r \,\mathrm d r$ give the area of a circle?

There's a method of computing the area of a circle by dividing it in concentric rings with infinitesimal width. Let $R$ be the radius of the circle and $r$ be the radius of the rings. The area of the ...
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1answer
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Green's theorem find circulation of vector field

$4.$ [$10$ Marks] Find the circulation of the vector field $$\vec F(x,y,z) = \langle x^{2018} -233x +y\cos x, 5x +\sin x +e^{2018y -233} \rangle$$ along the circle traced by $\vec r(t) = \langle 3\cos\...
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1answer
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Need to find where I went wrong on converting double integral to polar

Convert to polar form and solve $$\int^{1}_{0}\int_{0}^{\sqrt{2y-y^2}}(1-x^2-y^2)\text{ dx dy}$$ $x^2+y^2-2y=0$ $x^2+(y-1)^2=1$ $x=r\cos\theta$, $y=r\sin\theta+1$ $r^2=1$, $r=1$ $$\int^{\pi}_{0}...
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1answer
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A conversion of double integral to polar and evaluate, check

Problem 1 Convert to polar form and solve $$\int^{1}_{0}\int_{0}^{\sqrt{2y-y^2}}(1-x^2-y^2)\text{ dx dy}$$ $x^2+y^2-2y=0$ $x^2+(y-1)^2=1$ $x=rcos\theta$ $y=rsin\theta+1$ $r^2=1, r=1$ $$\int^{\...
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1answer
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Double integral conversions to Polar

Problem 1 Convert to polar form and solve $$\int^{2}_{0}\int_{0}^{\sqrt{2x-x^2}}((x-1)^2+y^2)^{5/2}\text{ dy dx}$$ $$x^2+y^2=2x, r^2=2rcos\theta , r=2cos\theta$$ $$\int^{\pi/2}_{0}\int^{2\cos\...
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1answer
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Convert integration to polar and solve

Evaluate the iterated integral $$\int_{-1}^1\int_0^{\sqrt{3+2y-y^2}}\cos\left(x^2+(y-1)^2\right)\,dy\,dx$$ Confused on how $y=0$ and $y=\sqrt{3+2y-y^2}$. Is this a typo or am I missing something?
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Closed form solution to $\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$

Is there closed form solution to this integral $$\int_0^{\pi/4}\frac{e^{ic\sqrt{1+ br^2(\theta)}}}{\sqrt{1+ br^2(\theta)}}\,d\theta$$ $r(\theta)=\frac{a}{\cos(\theta)}$ is radius vector from the ...
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Explicit parametrization for a 3-ellipse? A 4-ellipse?

I searched around but was unable to find anything. For the usual $2$-ellipse we have the parametrization $x(t) = a\cos(t)$ and $y(t) = b\sin(t)$ for $t\in [0,2\pi]$. Is there anything similar for ...
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1answer
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Multivariable Limit - Converting to Polar Coordinates

I am new to this concept, but I do know that, using Cartesian coordinates, if the limit is different for 2 different "routes", then it does not exist. I need to show that $$\lim_{(x,y)\to(0,0)}\frac{...
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2answers
72 views

Finding the angles of polar curves when calculating the area

I am trying to understand how to choose the angles when doing area calculations on polar curves. For example, to find the area inner loop of this limacon, $1+2\sin\theta$, I can identify four angles ...
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2answers
51 views

How would I prove $\frac{1}z=\operatorname{cis}(-\theta)$ given $z=\operatorname{cis}(\theta)$?

I know $\operatorname{cis}(-\theta)$ equals $\cos(\theta)-i\sin(\theta)$ and $\frac{1}z=\frac{1}\cos+i\frac{1}\sin=\sec+\frac{\csc}i$ (please, correct me if I'm wrong.) I just don't know how to ...
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1answer
78 views

Proving that a system of ODEs has a focus at the origin

Prove that the following system of ODEs has a focus at the origin. $$\begin{aligned} \dot{x} &= -x^3-y^3\\ \dot{y} &= x^3 \end{aligned}$$ Plotting the vector ...
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Finding the vertices of a regular n-sided polygon using its centroid

How do you find the vertices of a regular $n$-sided polygon using its centroid, with the knowledge that each vertice is distance $d$ from the centroid.
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1answer
58 views

Phase Portrait Using Polar Coordinates

I converted a system to polar coordinates and got: $$r'=r^2 \sin \theta \\ \theta'=r^2\cos\theta $$ Now I have to graph the phase portrait near the fixed point at (0,0) and don't know where to begin. ...
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1answer
41 views

Polar coordinates in dynamical systems

Using Polar coordinates, show that the system \begin{align} \dot x &= x -y -x(x^2 + y^2) \\ \dot y &= x +y -y(4x^2 + y^2) \end{align} gives $\dot r = r - r^3 (1+\frac34 \sin^2(2\theta))$. ...
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1answer
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Convert to polar coordinates, where 0 degrees means x = 0 and y > 0, increasing clockwise.

This isn't terribly different from from regular conversion to polar, but I'm having trouble adapting it. The results I'm looking for are in the range [0,360) that adhere to the following example data. ...
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2answers
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Deriving the Nautilus shell spiral equation

Suppose I need to estimate the cross section of a Nautilus shell, which is famously approximated by a logarithmic spiral, $r=ae^{b\theta}$. The cross-section of this spiral could be found by ...
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1answer
38 views

converting vector inside integral into polar coordinate

I am evaluating this function $A(z)$: $A(z) = \iiint v_x \frac{\partial f_o}{\partial v_x}(1-e^{-\frac{z}{\tau v_z}} ) dv_x dv_y dv_z$ $v$ is a vector in v-space. $\theta$ is the polar angle ...