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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7answers
2k views

Why does $r=cos\theta$ produce a circle?

I am trying to do a double integral over the following region in polar coordinates: I know that the limits of integration are: $$\theta=-\pi/2\quad to\quad \theta=\pi/2\\r=0\quad to\quad ...
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0answers
140 views

obtaining $y'$ from $r = 1 + \sin\theta$

Just want to check if this is the idea for this $$x = r\cos\theta$$ $$y = r\sin\theta$$ so now we substitute $$x = (1 + \sin\theta)\cos\theta$$ $$y = (1 + \sin\theta)\sin\theta$$ get the ...
1
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1answer
58 views

Determining the polar form for all n-th roots of unity.

By definition $ z \in\mathbb{C}$ is a n-th root of unity iff $z^n = 1$. My assignment is to (iv) List all n-th roots of unity in their polar form. You may use that there are only $n$ Elements with ...
0
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2answers
113 views

Ampersand curve in polar coordinates

I got an assignment to write a program which draws the ampersand curve. The equation of ampersand curve looks like this: $(y^2-x^2)(x-1)(2x-3)=4(x^2+y^2-2x)^2$ I was given an advice to convert this ...
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2answers
84 views

How to solve a complex equation $w^4 = \sqrt{3} -i$

$z = \sqrt{3} -i$ How do I solve a complex equation $w^4 = \sqrt{3} -i$ I know that I first have to rewrite z to polar format which I have done as $z = 2(cos (-π/6) + sin (-π/6))$ but I do not know ...
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1answer
446 views

Area of a square in polar coordinates?

I was attempting, for the exercise of it, to find the area of the a simple square with an infinite number of infinitesimal circle sectors. Let us say this square is $[5 x 5]$. Alas, it's been ...
2
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1answer
283 views

Use polar coordinates to find the volume of the given solid

Use polar coordinates to find the volume of the given solid bounded by the paraboloid $z=1+2x^2+2y^2$ and the plane $z=7$ in the first octant. I did it. Is that right ? $$\int_0^{\pi \over 2} ...
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1answer
55 views

confusing intergration

given $$r=4e^{3\theta} \space \space \space \space dr/d\theta=(3*4*e^{3\theta})$$ $$l=\int \sqrt(4e^{3\theta})^{2}+(3*4*e^{3\theta})^{2} \rightarrow $$ why does the integral $$ ...
3
votes
2answers
97 views

Find an estimation (using polar coordinates)

Consider the function $$ f(x,y):=\lVert x\rVert^{1-n}\ln(\lVert x\rVert)(\arctan(\lVert x-y\rVert))^{-\alpha},~~0<\alpha<n,~~n>1,~~(x,y)\in\Omega\times\Omega,~~~\Omega\subset\mathbb{R}^n $$ ...
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2answers
73 views

Laplace's equation in polar coords

Question: Suppose that the function u(r, $\phi$) satisfies Laplace’s equation for plane polar co-ordinates (r, $\phi$) i.e. $$ ∇^2u = \frac{1}{r} \frac{∂}{∂r}(\frac{r∂u}{∂r}) + ...
2
votes
1answer
50 views

How would you represent $y=(x-h)^2+k$ in polar coordinates?

I tried using $$x=r\cos(\theta)$$ and $$y=r\sin(\theta)$$ and ended up with $r\sin(\theta) = (r\cos(\theta)-h)^2 + k$ and wasn't sure how to proceed from there.
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0answers
21 views

Simple ways to create a separating plane given two points in a polar coordinate system?

I am currently working on sensor networks, where sensors are uniformly distributed in a polar coordinate system (maximum radius $R$ is set to $1$). A few of the sensors are placed equidistantly on a ...
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1answer
28 views

Problems with my work for double integral using polar coordinates

The question is as follows: My work goes like this: ∫∫R sin(x^2 + y^2) dA = ∫(θ from [0, 2π]) ∫(r from [1, 6]) sin(r^2) (r dr dθ) = [∫(θ from [0, 2π]) dθ] * [∫(r from [1, 6]) r sin(r^2) dr] = ...
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0answers
132 views

Finding an area using $r = 13\cos \theta$, $r = 6 + \cos \theta$

I have these: $$r = 13\cos \theta\quad r = 6 + \cos \theta$$ I am trying to find the area. Would anyone please help?
3
votes
4answers
202 views

How can the trefoil knot be expressed in polar coordinates?

From Wikipedia, the parametric equations for a trefoil knot are \begin{align*} x(t) &= \sin t + 2\sin 2t \\ y(t) &= \cos t - 2\cos 2t \\ z(t) &= -\sin 3t. \end{align*} I am only ...
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2answers
64 views

Convert double integral from cartesian coordiantes to polar coordiantes

I have the integral $$\int_{-3}^3 \int_0^\sqrt{9-x^2} (x^2 + y^2)^{3/2} {dy}{dx}$$ I cannot solve this in it's current form so I realize that the limit is a circle ${x^2} + {y^2} = 9$ using this I ...
2
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1answer
53 views

How to prove this ODE system is stable at origin?

A dynamical system in polar coordinates is: $$\Theta'=1, r'= r^2\sin(1/r), r>0, r'=0\mbox{ if }r =0.$$ How to show this is stable at origin? Intuitively, I really can't believe it because I ...
0
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0answers
19 views

Polar form of Bi-Quatenrion

I have a complex Quaternion(Bi-quaternion) and i want to convert that to Polar form (Euler). Let say we have Fourier transform of a ( bi-quaternion ) like, then how can we get a polar form ...
0
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0answers
38 views

Complex Integration using polar coordinates

Consider the complex variable $y=re^{j\phi}$ with $r\in(0,\infty)$, $\phi \in (-\pi,\pi)$, and the complex integral $$ I=\int\limits_\mathbb{C} {f(y)\log(f(y))dy} $$ Does the following ...
2
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1answer
43 views

smooth in polar but not in rectangular

Can anyone please give an example or two of functions on $\mathbb{R}^2$ which are smooth in the polar coordinates $(r,\theta)$ but not smooth in the Cartesian coordinates $(x,y)$? Thank you!
2
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2answers
39 views

Question about polar coordinates

I'm just learning about polar coordinates now, and I understand the basics pretty well, but I get confused at a particular part. I understand the following relations: $x = r\cos(\theta)$ $y = ...
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2answers
25 views

Polar Equation Conversion

Change the polar equation $\theta=\frac{\pi}{3}$ to rectangular coordinates. How would I go about this question? I've tried $x=r\cos\theta$ and $y=r\sin\theta$, but I can't figure out $r$ since ...
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0answers
52 views

Regions formed by polar coordinates in double integration.

I need to sketch the region of integration of the following double integral in the $xy$ plane: $$\int_0^{\pi/2}\int_0^{1/\cos\theta} f(r,\theta) \ dr \ d\theta,$$ where $f(r,\theta)= ...
2
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1answer
193 views

Jacobian for a Cartesian to Polar-Coordinate Transformation

I have a simple doubt about the Jacobian and substitutions of the variables in the integral. suppose I have substituted $x=r \cos\theta$ and $y=r \sin\theta$ in an integral to go from cartesian to ...
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0answers
37 views

Polar coordinates: fixing ratio between arc and radius

When drawing a coordinate system with fixed step size, the standard polar coordinates $$ x=r\cos(\theta), y=r\sin(\theta) $$ exhibits stretched pixels for large $r$. Ignoring the singularity in ...
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0answers
214 views

How to plot a stream function

This question relates to fluid mechanics and I have the components in polar coordinates. The components of the velocity field are; $$v_r= \frac{-kr}{z}$$ $$v_z= kz$$ $$v_\theta= 0$$ and I have ...
4
votes
1answer
57 views

Changing operator to polar coordinates

Let $$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$ be the Laplace operator on the $(x,y)$-plane. Consider the polar coordinates with $x=r\cos\theta$ and $y=r\sin\theta$. ...
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1answer
2k views

Velocity and acceleration of a particle in polar coordinates

I am asked to find the radial and transverse velocity and acceleration for a particle with polar coordinates $r=e^t$ and $\theta=t$ Therefore let $\underline{r}=\underline{\hat{r}}e^t$ and ...
0
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1answer
34 views

Path of an ellipse

A path is described by the position vector $\mathbf{r}$: $$\mathbf{r}=a\cos(\omega t)\mathbf{\hat{i}}+b\sin{\omega t}\mathbf{\hat{j}}$$ I am asked to show that the path is the ellipse in the form of: ...
1
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1answer
125 views

Rotation of 2D polar graph in a 3D space along some fixed axis?

Does there exist some systematic way of rotating a 2-D polar graph $r=f(\theta)$ around some axis in a 3D space? For example: $f(\theta)=cos(\theta)$ in 2-D looks like: If we want to rotate the ...
0
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2answers
730 views

Sketch $r=\cos(5 \theta)$? $r$ as a function of $\theta$ in cartesian coordinates

I think I have to plug in numbers into $\theta$ such as 0 and $\pi/6$. What kind of numbers should I plug in ? Sketch $r=\cos(5 \theta)$? $r$ as a function of $\theta$ in cartesian coordinates
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1answer
44 views

Heuristic approach to winding number

I'm working on problem 8.23 of Rudin's PMA, that is: Let $\gamma:[a,b]\to\mathbb C$ be a closed curve, $\gamma \in C^1([a,b])$ and $\gamma(t) \neq 0 \ \forall t\in [a,b]$. Show that $$\text ...
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1answer
431 views

Finding the centroid of a polar curve

I have absolutely no idea how to find the area centroid of this problem. I have been working at this one for ages but can't seem to get anywhere. Any first steps? How would one go about solving ...
0
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1answer
834 views

Can someone check my answer for this area between 2 polar curves question?

Find the area of the region that lies inside the circle $r = 1$ and outside the cardioid $r = 1-cos(\theta)$ I drew the graph and set it up like this: $$ \int_0^\pi \frac{1}{2} [ (1)^2 - ...
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0answers
112 views

Correct way to write the polar form of a complex number

What is the most correct way to write the polar form of a complex number? For example, given the complex number: $\dfrac{\sqrt{3}}{2} + \dfrac{1}{2}i$ I would write the polar form as follows: ...
2
votes
0answers
108 views

Polar Coordinates for Multivariate Limits With Three Variables

When working on limits with two variables, $f(x,y)$, I like to convert the problem to polar coordinates a lot of the time, by changing the question from $$\displaystyle\lim_{(x,y)\to (0,0)}f(x,y)$$ to ...
0
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1answer
51 views

Representation of differentials in Polar Coordinates

We define polar coordinates in $\mathbb{R}^{n}$\ $\{ 0\}$ by $x=ry$, where $r=|x|>0$ and $y \in \partial B(0,1)$ is a point on the unit sphere. In the coordinates, Lebesgue measure has the ...
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0answers
94 views

Ambiguity in converting from cartesian to polar coordinates

Just started looking into complex numbers in "The Art of Electronics" book. When converting from cartesian $(x,y)$ coordinates to polar (r,$\theta$), the conversions are carried out according to the ...
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0answers
41 views

what are the possible solutions to this equation?

I'm trying to find some angles for my characteristic equation , I need to know the roots or possible answers to cosine equation $$1-\cos(u)\cdot\cosh(w)=0,$$ and $$u=\sqrt{\lambda_1}\cdot L.$$ ...
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0answers
99 views

Polar equation of perimeter of half ellipse

x = Cx + a * cos(ang); y = Cy + b * sin(ang); Cx, Cy are cords of center. ...
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1answer
130 views

Conversion to polar equation

I would to know when asked to convert an equation to polar what it means.For example $ x^2+x+y^2-2y=0 $ My understanding so far tells me I need to derive an equation in form of: $$ r^2=x^2+y^2$$ ...
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2answers
206 views

Image of a closed curve under $w=z^2$.

I have the curve: $$r=2(1+ \cos \theta), \ \theta \in [0,2\pi)$$ in polar coordinates on the complex $z$ plane, and I want to find the image of this curve under the square function $w=f(z)=z^2$. ...
0
votes
1answer
456 views

Hyperbola in polar coordinates, what's wrong?

I read that the equation of a conic in polar coordinates is $$r=\frac{l}{1+e\cos \theta}.$$ But when I try to reduce the hyperbola $$x^2 - y^2 =1$$ to that form by setting $x=r\cos \theta $, $y=r ...
0
votes
3answers
467 views

Pushforward of a vector field

Can someone help me with that ? We define $\phi:=(\phi^1,\phi^2):\Omega\subset\mathbb{R}^2\to\phi(\Omega)$ with $\Omega$ such that $\phi$ is a diffeomorphism by ...
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2answers
252 views

Evaluating the area in the polar coordinates

So the problem asked me to find the area of the region that lies inside both of the circles $$r=2sin\theta, \quad r=sin\theta +cos\theta $$ I know that $r=2sin\theta$ is $x^2+(y-1)^2=1,$but ...
0
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1answer
50 views

change of variables while integrating

Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...
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1answer
72 views

How to use $dz=d[r(t)(\cos t + i\sin t)]$ as a change of coordinates?

This notation comes in handy for some path integrals, but I don't know yet how this is calculated. Is it simply a change of coordinates? Is this correct: $$z=r(t)(\cos t+i\sin t) \quad ...
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2answers
330 views

Equation of circular sine waves in the water

I have to write the equation of a sine wave expanding circularly from a point $P_0=(x_0,y_0)$. The wave has the form $\eta(\rho)=A\sin(\omega\rho)$ where $\rho$ is the distance from the point $P_0$. ...
0
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2answers
133 views

need help finding the coordinates of AB

Find AB if the coordinate of A is -5 and the coordinate of B is 17. i have been out of school for over 20 years and have little to no memory of this process. i examined my daughter's book and there is ...
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1answer
138 views

Software for drawing two-variable functions in polar coordinates

I am in difficulty of finding a software for drawing two-variable functions in polar coordinates. Could someone introduce useful software for me? Thanks in advance. For example $$ f(r, ...