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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3
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6answers
389 views

Simple partial differentiation $x = r\cos\theta$ and $y = r\sin\theta$

If \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 ...
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4answers
350 views

Find the length of a curve

Find the length of the following curve: $r(t) = e^{-t} \sin(t)+e^{-t}\sin(t) i$ for $0 \leq t\leq 1$. Any ideas?
2
votes
0answers
239 views

Drum membrane wave equation general solution (non-symmetrical)

I think I am stuck in solving this problem. It involves a wave equation in a circular membrane, so polar coordinates must be used: $u_{tt}=c²(u_{rr}+{1\over r}u_r+{1\over r²}u_{\theta\theta})$, ...
1
vote
0answers
123 views

Circle-Circle intersection coordinate system

Consider two points in the 2D Euclidean plane, the origin $0$ and $x$. One can define a co-ordinate system such that for any point $y$ in the plane, $y$ is parametrized by its distance from $0$, call ...
0
votes
1answer
996 views

Numerical Integration of a Gaussian Distribution in Polar Coordinates

I want to numerically evaluate a 2D-integral of a specific probability distribution over some given area (I use MATLAB so all the code below is MATLAB code). I broke down the problem so that it ...
1
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4answers
5k views

Find the area of the region inside the limaçon

I'm struggling to figure out the answer to this: Find the area of the region inside the limaçon, $r=3 + \sin(\theta)$ Could someone please help me out?
5
votes
1answer
1k 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 ...
4
votes
1answer
895 views

Bézier approximation of archimedes spiral?

As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ...
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2answers
2k views

Del operator in 2D polar coordinates

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: ...
0
votes
2answers
222 views

Length of a plane curve in polar coordinate

Consider the plane curve $\gamma$ in polar coordinates: $$ r=r_0+e^{\lambda\theta}, \quad \theta_1 \le \theta \le \theta_2, $$ where $r_0,\lambda,\theta_1>0$. Is it possible to compute explicitly ...
1
vote
2answers
7k views

area between polar equation $r = \sin\theta$ and $r = \cos\theta$

Below is the exact question and answer from my textbook: Find the area of the region enclosed between the two curves $C_{1}$ and $C_{2}$ where $C_{1}$ has the polar equation $r = \sin\theta$ and ...
3
votes
4answers
4k views

find all points for intersection between 2 polar equations

I stumped at one of the exercise in my multivariable calculus textbook. I try to search online but I can't seem to search on how answer no 3 and 4 below is derived. I also plot both of polar ...
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1answer
5k views

Find the equation in polar coordinate form for a straight line through the points with polar coordinates (4,0) and (4,π/3).

Find the equation in polar coordinate form for a straight line through the points with polar coordinates $(4,0)$ and $(4,π/3)$. Here's my steps: 1.Write the two points in cartesian coordinates: the ...
4
votes
2answers
900 views

Find the area enclosed by the loop $r=2(1-\sin\theta)\sqrt{\cos\theta}$

The diagram shows a sketch of the loop whose polar equation is $$r=2(1-\sin\theta),\qquad -\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$$ a)Show that the area enclosed by the loop is 16/3. ...
0
votes
1answer
2k views

Polar equation of cartesian $y = 1 + 3x$

I have no idea at all what to do on this I got $$\cos^{-1} \left(\frac{r\sin \theta+1}{3} \right) = \theta$$ Which can be $$\cos^{-1} \left(\frac{r\sin \left(\cos^{-1} \left(\frac{r\sin ...
0
votes
2answers
955 views

Polar equation of $y = 2$

Maybe I do not understand what is going on here but I cannot get the right answer. $$y = 2 $$ $$y^2 + x^2 = r^2$$ $$4 + 0^2 = r^2$$ $$ r = 2$$ $$y = r \sin \theta$$ $$1 = \sin \theta$$ $$\theta ...
2
votes
1answer
75 views

gradient in polar coordinate by changing gradient in Cartesian coordinate

I'm tried to do following and I can't see what went wrong. $$\begin{bmatrix} \hat r\\ \hat \theta \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta ...
0
votes
2answers
927 views

Single variable integral to polar coordinates?

I took calculus about 2 semester ago, and I'm trying to brush up on polar coordinates. I integrated $-x^2+3$ from $x = -\sqrt{3}$ to $\sqrt{3}$ and I got $6.93$ Now I tried to convert it to polar ...
1
vote
1answer
349 views

Maximum point of a polar function

I have a curve C with polar equation $$r^2 = a^2\cos{2\theta} $$ And I am looking to find the length $x$ when $r=max$ Judging from the equation: $$r = \sqrt{a^2\cos{2\theta}} $$ R will be maximum ...
1
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1answer
378 views

Find the area of the region determined by two curves

Find the area of the region $R$ given by two curves. So the region $R$ describes the area that is common between the two curves: $$\begin{align*} \text{Function 1: } r&= 2\sin(\theta)\\ ...
5
votes
2answers
125 views

Why does it always take n numbers to characterize a point in n-dimensional space (or does it)?

I don't know if this is obvious and a dumb question or not, but, here we go. To characterize a point in 2-d space we can use standard $x,y$ coordinates or we can use polar coordinates. There are ...
5
votes
2answers
1k views

Is $r=2\cos(\theta)$ a one-petal polar function?

I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
3
votes
1answer
2k views

How to “rotate” a polar equation?

Take a simple polar equation like r = θ/2 that graphs out to: But, how would I achieve a rotation of the light-grey plot in this image (roughly 135 degrees)? Is ...
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vote
1answer
359 views

Fitting of Closed Curve in the Polar Coordinate.

I know how to fit a curve when given some data points in the cartesian coordinate. Recently, I encountered a model that needs to fit a closed curve in the polar coordinate. I'm thinking of deducing a ...
3
votes
4answers
2k views

Polar equation of a circle

A very long time ago in algebra/trig class we did polar equation of a circle where $r = 2a\cos\theta + 2b\sin\theta$ Now I forgot how to derive this. So I tried using the standard form of a circle. ...
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2answers
5k views

Set up double integral of ellipse in polar coordinates?

How do you set up a double integral for an ellipse in polar coordinates without using Jacobian or Greens Theorem? I can't seem to figure out what (or if) the limits of r can possible be. $x = ...
0
votes
3answers
2k views

Argand Diagram - Quadrants help

I forgot the rules of adding angles when it comes to argand diagrams. In the first quadrant, you add 90 degrees to whatever angle you get, what about Q2 Q3 Q4 ? This picture will explain what i mean ...
2
votes
1answer
1k views

Find the Cartesian equation corresponding to $r = \frac{5}{3-2\cos(\theta)}$

Find the Cartesian equation corresponding to $r = \frac{5}{3-2\cos(\theta)}$ I got it into the form: $(x^2 + y^2)(3-2x)^2 = 25$ and can see that maybe the equation of a circle will appear, but ...
9
votes
4answers
4k views

Simple proof of integration in polar coordinates?

In every example I saw of integration in polar coordinates the Jacobian determinant is used, not that i have a problem with the Jacobian, but I wondered if there's a simpler way to show this which ...
2
votes
1answer
269 views

Find volume of a revolved solid by integrating wedges.

So, lets say that I wanted to find the volume of the solid formed by rotating the area between $f(x)=\sqrt{1-x^2}, 0<x<1$ and the $x$ axis around the $y$ axis. (This example is simply a ...
2
votes
2answers
144 views

Integral variable substitution using Hausdorff measure

Suppose we have positive density $q$ with "good" qualities (continuity, etc..). I need to calculate this integral: $$\int_B q(\textbf{z}) d \textbf{z},\ \textbf{z} \in \mathbb{R}^d,$$ where $B \subset ...
1
vote
3answers
484 views

Expressing $e^z$ where $z=a+bi$ in polar form.

I am reading a passage of text that states: "We can use the fact that $e^{a+bi}=e^a(\cos b+i\sin b)$ has polar form $\left<e^a,b \right>$ to verify that complex exponentials have various ...
1
vote
1answer
218 views

Complex Numbers and polar form

I am given the following information: $$x[n]= s^n,\qquad n=0,\pm 1,\pm 2,\ldots$$ where $s=\sigma + j\omega = re^{i\theta}$ is a complex number in general. I was wondering how the following is ...
0
votes
1answer
3k views

How to write parametric equations for a given polar equation?

I'm doing an extra credit problem for math, we haven't learned too much on this topic. The instructions are: Write parametric equations for the given polar equation. The problem is: $r = ...
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1answer
1k views

Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation.

Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation. Let $t= \theta$, $r=5$, $x=r\cos t$, $y=r\sin t$. I began with $x=5\cos t-5=5(\cos t-1)$ and $y=5\sin t$. Is that ...
0
votes
1answer
341 views

Finding a geodesic on a plane using polar coordinates

This is from my homework on PDE. I need to find a geodesic on a plane using polar coordinates. Now, I know $dl^2 = x^2+y^2$ hence $l=\int \sqrt{dx^2+dy^2}$, but I get stuck while converting ...
0
votes
2answers
137 views

Laplacian in polar coordinates

I am stuck with an exercise that requires me to find the Laplacian $\Delta u=(D_x^2u+D_y^2u)$ of a 2d-function $u$ in polar coordinates (in the standard Euclidean plane). I found the following ...
2
votes
1answer
140 views

Explain why for $r=1-a \cos^2(3\theta)$ the leaves have the same size only in the case $a=1$ and $a=2$

Please explain why for $r=1-a\cos^2(3\theta)$ the leaves have the same size only in the case $a=1$ and $a=2$. Does anyone have an answer to this please?
0
votes
1answer
525 views

Transform integral into polar coordinates

At university we are given a voluntary hand in in the use of maple/matlab, in that regard I have a double integral I am in dire need to compute, using first cartesian then polarcoordinates. ...
2
votes
1answer
994 views

How do I change a 3D cartesian equation into a polar equation?

I know how to change 2D cartesian equations into polar equations, however I'm having some difficulty with a 3D equation. I am trying to take the cartesian equation x^2+(.75y+4)^2+(z+3)^2=20 and turn ...
0
votes
1answer
63 views

Find complex $z$ such that $z$ has the largest possible real part, and satisfies: $z^7 = -18-18i$

Find complex $z$ such that $z$ has the largest possible real part, and satisfies the equation: $z^7 = -18 -18i$ So, the 7th roots of $z = 18\sqrt{2}e^{i\frac{\frac{\pi}{4} + 2\pi k}{7}}$ ...
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2answers
62 views

Find the Cartestian form of $6 - 7i$ rotated anticlockwise through $\frac{3\pi}{4}$ about the origin

Find the cartestian form of $6 - 7i$ rotated anticlockwise through $\frac{3\pi}{4}$ about the origin I realize that I am going to be doing something like: $\sqrt{85}e^{i\alpha}.e^{i\frac{3\pi}{4}}$ ...
2
votes
1answer
2k views

Double Integral, Change of Variables to Polar Coordinates

Quick question on Polar Coordinates. When evaluating the double integral and changing variables, I'm not sure if the limits are correct. The question is as follows: Evaluate $$\int\!\!\!\int_D ...
0
votes
3answers
235 views

Write $\cos(9x)$ in terms of powers of $\cos(x)$ [duplicate]

Possible Duplicate: How to expand $\cos nx$ with $\cos x$? Write $\cos(9x)$ in terms of powers of $\cos(x)$ I realize I could solve this by using De Moivre's and binomial expansion: ...
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vote
1answer
127 views

Given an exact velocity and a “velocity range”, what is the relative velocity range?

I'm trying to calculate the relative velocity ($V_R$) between an exact velocity ($V_0$) and a velocity range ($V_1$). The exact velocity ($V_0$) is represented simply by ($course$, $speed$). The ...
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2answers
4k views

Difficult conversion from polar equation to rectangular equation.

How do we convert this into rectangular equation? $r=5\theta$
0
votes
2answers
210 views

How do we get the rectangular form of this?

I know if $\sqrt{x^2+y^2} = x$, then the polar equation of this is $r=cos\theta$ So,how to get the rectangular form of this polar equation, is it complicate: $r=cos(10\theta)$
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votes
1answer
1k 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, ...
2
votes
2answers
133 views

Polar Coordinates

It's been ages since i did any coordinate conversions, and typically i have these two which i just can't manage to solve by myself. I want to express the circle $x^{2}+y^{2}<4, x<0 $ The Area: ...
1
vote
1answer
276 views

Help needed with partial derivatives and polar coordinates, missing term.

I have a missing $\frac{1}{r}\partial_r$ -term (notice the question mark) but cannot see why, could someone hint where I am doing mistake.