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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Polar co-ordinates dr/dtheta

How can you visualise what is the curve doing by calculating Dr/dtheta in polar co-ordinates form. Also, what will it mean for Dr/dtheta to be zero? Thank you.
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1answer
48 views

Contravariant vector example with polar coordinates

My book gives me this definition for contravariant vector: Let an n-tuple of real numbers $a^1,a^2, \dots, a^n$ be associated with a point P of an n-dimensional Riemannian space with coordinates ...
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2answers
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Converting Polar Equation to Cartesian Equation: general form solution

I'm trying to find the Cartesian equivalent of the general equation $$r=a\cos(q\theta) + c; q\in\mathbb Q, a\gt c \in\mathbb R$$ if it exists. My memory of calc is a bit hazy, and I haven't been able ...
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35 views

How to Trace a Real-Life Flower Using Polar Equations?

Here is the flower I'm trying to trace: $\hskip2cm$ How can I trace this flower using polar equations? I currently have the formulas \begin{align} r_{1}&=1.75\sin(10\,\theta + 18) +3\\ ...
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1answer
30 views

Polar equation for a k-leaf rose: is it possible to define an inner radius?

Is it possible to define a polar equation for a k-leaf rose with an inner radius for a k-leaf rose (as in this image)? I'm familiar with the general equation for a k-leaf rose $$r = \cos(k*\theta)$$ ...
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1answer
35 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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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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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 ...
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1answer
35 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 ...
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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 ...
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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 ...
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1answer
74 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 ...
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1answer
33 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: And then goes on to proving: The first inequality chain ...
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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 ...
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1answer
44 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),\\ ...
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67 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: ...
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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 ...
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3answers
120 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 ...
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2answers
65 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 ...
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1answer
22 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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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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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 ...
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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
46 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
37 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
28 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 ...
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1answer
41 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
26 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
18 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 ...
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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?
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1answer
31 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
21 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 ...
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1answer
48 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 ...
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1answer
11 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 ...
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1answer
12 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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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 ...
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1answer
19 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
98 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 ...
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89 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 ...
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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
35 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
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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
36 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
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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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256 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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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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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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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
48 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) $\dfrac{\sqrt{3}+\iota}{2}$ ...