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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1answer
44 views

Convert $y^2 = 4(x + 1)$ to a polar equation

I'm trying to convert the rectangular cartesian equation $$ y^2 = 4(x + 1) $$ to a polar equation. After replacing $y = r \sin \theta$ and $x = r \cos \theta$, I get $$ r^2 \sin^2 \theta = 4(r \cos ...
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0answers
21 views

Cylindrical coordinate derivative of a vector field.

Considering the following identity transformation in cylindrical coordinate: $$\mathbf{v}(R,\theta,z)=R\;\mathbf{e}_{R}+\theta\;\mathbf{e}_{\theta}+z\;\mathbf{e}_{z} $$ Taking its derivative ...
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0answers
20 views

How to compute this integrale $\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2} } dx$

I would like to calculate the following integral $$I(a,y)=\int_{\mathbb R^3} e^{-i\left<x,y\right>} e^{-a\| x\|} \| x\|^{\frac{5}{2}} dx, \quad a>0, y\in \mathbb R^3 .$$ Here's what I did: In ...
3
votes
1answer
38 views

Polar coordinates integration

Compute the following integrals over $R$ $f(x,y)\,dx\,dy$ over the area $R$ where: $f(x, y) = x$ and $R$ is given by $0 ≤ r ≤ \cos θ$ and $f(x, y) = x$. I understand polar coordinates is probably ...
0
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0answers
38 views

World To Screen Game math

I have a Coordinate system, I have my XYZ, pX,pY,pZ and the other player, eX, eY, eZ and I want the Pitch and YAW First the YAW: I first take VectorX = eX - pX VectorZ = eZ - pZ then I ...
1
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0answers
14 views

Hint for setting up this surface integral

\begin{equation} \iint_S z+x^2y \,\, dS \end{equation} Where S is the part of the cylinder $y^2+z^2=1$ that lies between the planes $x=0$ and $x=3$ in the first octant. I tried to convert to ...
2
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1answer
77 views

The Plot of a Leaf

Motivation Recently, when I was doing some searches for some syntax in the help pages of my Computer Algebra System (CAS), accidentally, I found this parametric curve in polar coordinates ...
2
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1answer
62 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 ...
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0answers
27 views

Polar form of generalized superellipse

I am looking for the polar form of the generalized superellipse: $$ \left|\frac{x}{a}\right|^{n_2}+\left|\frac{y}{b}\right|^{n_3}=1 $$ where $a$ and $b$ are the semi major and semi-minor axes. I have ...
2
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1answer
46 views

Solve the double integral $\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)dxdy\:$ [closed]

$$\int _{-1}^1\int _{-\sqrt{1-4y^2}}^{\sqrt{1-4y^2}}\left(3y^2-2+2yx^2\right)\,dx\,dy.$$ I think you need to be solved by the transition to polar coordinates: \begin{cases} x=r\cos(\phi),\\ ...
0
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1answer
72 views

Simultaneous equations in polar coordinates

I want to find the intersections of pairs of curves in polar coodinates. As an example, I have three circles with different offsets in a plane which you can see here. The offsets are: ...
1
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1answer
27 views

Find limits of integral for plane polar co-ordinates question

Use plane polar co-ordinates or otherwise to evaluate the integral $$\int\int_D^\ \frac{x^2-y^2}{x^2+y^2} dA$$ where D is the part of the x,y plane bounded by the parabola $y^2=4(1-x)$ and the ...
8
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3answers
134 views

how to integrate this $\int_0^{\infty} r^2 e^{\frac{-r^2}{2}} \, dr$?

What am I doing wrong when integrating this? $$\int_0^{\infty} r^2 e^{\frac{-r^2}{2}} \, dr$$ I used integration by parts and set $u=r^2$ and $dv=e^{\frac{-r^2}{2}}dr$ and I get ...
2
votes
2answers
115 views

The function $f(r,\theta)=(r\cos\theta,r\sin\theta).$

Consider the function $f:\mathbb{R}^{2}\rightarrow\mathbb{R^2}$ given by $$f(r,\theta)=(r\cos\theta,r\sin\theta)$$ I like to show that $f$ is one-to-one in some neighborhood of any non zero point ...
1
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1answer
26 views

${\int\int\int}_B dxdydz$ where $B$ is the region delimited by $x²+y²+z² = 4$ and $x²+y²=3z$

Take the following integral over the specified region: ${\int\int\int}_B dxdydz$ where $B$ is the region delimited by $x²+y²+z² = 4$ and $x²+y²=3z$ (i'm answering my own question because I was ...
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0answers
23 views

Integration problem in polar

How to integrate double integral $$\int_{0}^{\infty}\!\int_{0}^{2\pi}\ \frac{1}{2}\left(\frac{\partial}{\partial x}-\frac{\partial}{\partial y}\right)g_m \bar{g_n} , d\theta dr$$ where ...
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0answers
35 views

Ways of representing “half-way” between applying a homeomorphism?

Something I am particularly interested in is finding a potential way to create an animation illustrating smoothly how points in one 2D space map to another. In particular, I would like to show ...
3
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0answers
34 views

Green's Theorem with respect to a given polar region.

Using Green's Theorem, compute the counterclockwise circulation $I$ of $\vec{F}=\langle-\sqrt{x^2+y^2},\sqrt{x^2+y^2}\rangle$ around the region defined by the polar coordinate inequalities $7 ...
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2answers
41 views

Evaluating area D using polar coordinates

Let $D$ be the region in the xy-plane bounded on the left by the line $x=2$ and on the right by the circle $x^2 + y^2 = 16$. Evaluate $$\iint (x^2 + y^2)^{-3/2}dA$$
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1answer
49 views

Line Integrals - Calculus

I have a problem asking me to find $\int_C \textbf{f} \cdot d\textbf{r}$ where $\textbf{f}$ = $(\sin y,x\cos y)$, and the curve $C$ is any closed circle. I'm struggling with this, so far I have found ...
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1answer
57 views

How to find the domain of each petal in a Polar graph?

Given the equation $r=4\cos(3\theta)$, how can I find the domain of each petal? Help!
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1answer
49 views

Non-simultaneous intersections of $r = 4\cos\theta+1$ and $r = 2\cos\theta+1$

$$r = 4\cos\theta+1$$ $$r = 2\cos\theta+1$$ This system has simultaneous solutions at $(1, \frac\pi2)$ and $(1, \frac{3\pi}2)$. But looking at the graph, there are non-simultaneous intersections at ...
2
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1answer
71 views

Applied Mathematics: Spherical Polar Coordinates and Newton's Second Law

I've been attempting this question but can't seem to find a solution. Question: A particle of mass $m$ moves under the influence of a force which, in spherical polar coordinates, only acts in the ...
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1answer
49 views

Find the points on the given curve where the tangent line is horizontal or vertical.

Please help! I don't know how else to do this question. Thank you!!
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1answer
19 views

Max and minimum value that function $x*e^{x^2+y^2}$ can take on D

So I have to find the maximum and minimum value that the function $~xe^{x^2+y^2}~$ can take on: $$ D = \bigl\{(x,y) :\, 9 \leq x^2 + y^2 \le 16,~ y \geq 0\bigr\} $$ I've converted the Cartesian ...
0
votes
2answers
24 views

Exponential to polar form

I have exponential form $$ je^{-j\pi/2} $$, where $j = \sqrt{-1}$ I want to convert this to polar form $$j(\cos\pi/2 + j \sin \pi/2)$$ is it correct?
0
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1answer
33 views

Polar coordinates: what is the area of the region inside the inner loop of $r = \cos (\theta) - \frac12$?

I'm struggling plotting $r = \cos (\theta) - \frac12$. I've done it in Cartesian but I can't quite get in polar coordinates. I know it is supposed to be a loop but how do I get it? Being that I have ...
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1answer
26 views

Change of variables - Double integrals

I have trouble understanding how the limits work regarding polar coordinates in a double integral. For example, say if I had the equation $$(x-2)^2 + y^2 = 1.$$ This is a circle centred at (2,0) with ...
0
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1answer
79 views

Evaluate the double integral by changing to polar coordinates

I experience some difficulty with converting to polar coordinates in integrals. So the question I'm struggling with is Evaluate the double integral $$\int\int x^{6}y\, dA$$ where $D$ is the top ...
1
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1answer
12 views

What PC programs or iPad applications are there which allow you to plot cylindrical/spherical polar graphs?

I've been trying to get my head around the use of cylindrical and spherical polars to plot graphs. I feel that the easiest way to do this would be to try plotting some, but I'm struggling to find a ...
0
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1answer
14 views

How would you use cylindrical polar coordinates to find the area of a cone (and why does my method not work?

The following question was recently asked in a lecture: Using cylindrical polar coordinates find the area of the curved surface of a cone of height $h$ and radius $a$. My attempt to do this was ...
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0answers
467 views

How to solve this integral to find the exact length of an equation in the polar plane?

I hope it is only because it's late and I've been studying for a calculus exam for several hours, but I cannot see how to solve this integral. The problem states: Find the exact length of the ...
0
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1answer
25 views

Complex Numbers Midpoint of Roots of Unity

A = $\sqrt{2}e^{i(\frac{7\pi}{12})}$ B = $\sqrt{2}e^{i(\frac{11\pi}{12})}$ Express the midpoint M of AB in the form $a + bi$ (a,b in simplified surd form) I know M = (A+B)/2 but I cant find A+B in ...
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2answers
804 views

Find the area of the region that lies inside the first curve and outside the second curve. $r = 10 \cos\theta,\ r = 5$

I am not sure of my answer. In the figure, $r=10 \cos\theta$ is a circle that doesn't look like a circle. The area of $r=5$ is $\pi r^2 = 25 \pi$. You remove the area from $-\pi/3$ to $\pi/3$ of ...
2
votes
2answers
90 views

How to get the area between these $2$ functions?

I have a function: $(a)$ $r = 4\cos(2\theta)$ $(b)$ $r = 4\sin(2\theta)$. I need at least a set up for the integral that will yield the area inside the rose (a) but outside the rose $(b).$ I cant ...
70
votes
3answers
1k views

Cardioid in coffee mug?

I've been learning about polar curves in my Calc class and the other day I saw this suspiciously $r=1-\cos \theta$ looking thing in my coffee cup (well actually $r=1-\sin \theta$ if we're being ...
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4answers
42 views

Does an integral of a polar function from $0$ to infinity have to diverge?

This is more of a theoretical question, but I was curious if a polar equation automatically diverges as it goes to infinity? After all, the area will just be the area in the polar graph added to ...
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3answers
49 views

Evaluate iterated integral by changing to polar coordinates

$$\int_0^{1/2}\int_0^{\sqrt{1-x^2}}xy\sqrt{x^2+y^2}\,dy\,dx$$ $x^2+y^2=r^2$ $$\int\int_0 r^3\cos\theta \sin\theta|r|\,dr\,d\theta$$ I don't know what $r =$ at line $x = 1/2$. I don't know value of ...
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3answers
39 views

Converting Polar Equation to Cartesian Equation problem

So I have 1. $$\frac{r}{3\tan \theta} = \sin \theta$$ 2. $$r=3\cos \theta$$ What would be the Cartesian equation???
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4answers
30 views

Polar Equation to Cartesian Equation.

Find the cartesian equation of the circle with polar equation $r=2a\cos \theta$ My attempt, Since $\cos \theta=\frac{x}{r}$ So, $r=2a(\frac{x}{r})$ I don't know how to proceed.
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2answers
357 views

Spiral of Archimedes area and sketch in polar coordinates

This is an exercise from Apostol's Calculus, Volume 1. It asks us to sketch the graph in polar coordinates and find the area of the radial set for the function: $$f(\theta) = \theta$$ On the interva ...
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0answers
36 views

Rewrite the equation of a conic in cartesian coordinates

Consider the equation for a conic in polar co-ordinates $(r,\theta)$ $$r = \frac{k}{1 - e\cos(\theta)} \qquad \qquad (1)$$ in the case where $k > 0$ and $e > 1$. Show that ...
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0answers
15 views

How would you express this integral in cylindrical polar coordinates?

How would you express the integral \begin{gather*} \int_{0}^{1}\int _{0}^{\sqrt{1-x^{2}}}\int_{0}^{1-x^{2}-y^{2}} e^{z} \ dz \ dy \ dx \end{gather*} In cylindrical polar coordinates, would it be ...
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0answers
22 views

How to read a 3D Polar Plot?

r = 1 - sin[n*theta] I'm an integral calculus student and I'm trying to interpret 3D polar graphs. I know that r is the amplitude for polar coordinates and n is a scalar for theta which dictates the ...
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3answers
49 views

The value of the cubic root of $-i$

So this was the question given to us. $\left(\iota=\sqrt{-1}\right)$ Value(s) of $\left(-\iota\right)^{\dfrac{1}{3}}$ are (A) $\dfrac{\sqrt{3}-\iota}{2}$ (B) $\dfrac{\sqrt{3}+\iota}{2}$ ...
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0answers
9 views

Partially differentiating interrelated equations

I apologize if this question is too simple or if I am not following guidelines in any way. This is my first question, so constructive criticism is appreciated. Anyway, I am given Z = x^2 +2y^2 x = ...
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1answer
18 views

Finding the polar form of a complex number

I have the following complex numbers : -3,18 +4,19i I can calculate $r=\sqrt{a^2+b^2}$ Which gives r=5,26 now I know that cos $\theta = \frac{a}{r}$ gives $\theta=127,20$ degrees But when I do ...
0
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2answers
73 views

Double integral with Polar coordinates - hard example

Calculate using polar coordinates: $$\iint_{D}^{} (x^2+y^2)^\frac{1}5 \ dx \ dy $$ where D is the region inside the circle with radius 1. Working: D: $ \ x^2+y^2=1 \\ $ so $ 0 \leq r \leq 1 \ \ , $ ...
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0answers
31 views

Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
1
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1answer
29 views

Finding the area of a polar region

I am trying to find the area inside the curve $$ r = 2 + \sin2\Theta + \cos3\Theta .$$ It's a very weird looking function after graphing, and I'm not quite sure how I'm supposed to proceed. There's ...