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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2answers
106 views

$x² + y² +6y = 7$ to polar coordinates

How do i come from $M1$ to the polar coordinates? $$M1 \qquad x^2 + y^2 +6y = 7$$ I started with: $$r^2 * (\cos^2\varphi + \sin^2\varphi)+ 6r\sin\varphi = 7$$ because: $$(\cos^2\varphi + ...
1
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1answer
122 views

translate coordinates on circle to percentage?

I'm coming more from a programming point of view but the question is pure math. The only strange thing, I guess, is that the coordinate system is like this: ...
27
votes
2answers
6k views

Cannabis Equation

How can an equation for the following curve be derived? $$r=(1+0.9 \cos(8 \theta)) (1+0.1 \cos(24 \theta)) (0.9+0.1 \cos(200 \theta)) (1+\sin(\theta))$$ (From WolframAlpha)
1
vote
1answer
91 views

Write function from polar to rectangular coordinates.

I need to write this functions in rectangular coordinates: $$f(r,\theta)=r^{2k+5}\cos5\theta$$ $$g(r,\theta)=r^{2k+5}\cos5\theta$$ Of course the radius is very easy to convert to $x$ and $y$. The ...
2
votes
1answer
119 views

Bounding an integral over a surface (using polar coordinates?)

Suppose $S$ is smooth $n-1$-dimensional closed and bounded (compact) hypersurface in $\mathbb{R}^n$. Suppose for simplicity that $S$ is the boundary of a Lipschitz domain, for example. Whatever makes ...
0
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2answers
628 views

What is the area outside of $r=1$ and inside $r=2 \cos(3 \theta)$?

What is the area inside the polar curve $r = 2 \cos(3 \theta)$ but outside of the circle given by the polar equation $r = 1$? A picture of the polar curve is at ...
2
votes
2answers
1k views

Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Need help solving 11.bi), A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You ...
1
vote
1answer
2k views

Find the area of the region R inside the circle $ r=2 \cos \theta$ and outside the cardioid $r=2− 2 \cos \theta$ .

I got $\theta\pi/3$ and $5\pi/3 $ and then the area I got was $-4\sqrt3-(8\pi)/3$ The area is not right, I used the area equation that takes integral of $1/2(f(\theta)^2-g(\theta)^2)$ from $\pi/3$ ...
0
votes
3answers
465 views

Changing from Cartesian coordinates to Polar coordinates

Rewrite the iterated integral $$\int_0^1 \int_0^{\sqrt{2y - y^2}} (1 - x^2 - y^2)\,dx\,dy$$ in polar coordinate form. Do not evaluate the integral. Here is my answer: ...
0
votes
2answers
145 views

Find the area enclosed by the curve $x=t-\sin t $, $y=1-\cos t$ from $0$ to $2\pi$

Find the area enclosed by the curve $x=t-\sin t $, $y=1-\cos t$ from $0$ to $2\pi$ and the x-axis Not sure how to execute. is it just that $.5\int_{0}^{2\pi}x^2+y^2$ ? and not sure how to account ...
2
votes
1answer
80 views

Problems with integrals, polar coordinates.

I am having problems parameterizing these integrals: $$\int_A{\frac{x}{1+x^2+y^2}}\mathrm{d}x\mathrm{d}y$$ for $A = \mathbb R^2 \bigcap \,\{y \ge 0\}$ and the volume of $M = \{(x, y, z) \in \mathbb ...
0
votes
2answers
89 views

Find the area inside a polar curve

I feel a bit silly asking this question as it is no doubt relatively simple, but it has been bugging me. Given the polar curve described by $r^2 = cos(2\theta)$, find the area inside the curve. My ...
2
votes
7answers
3k 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 ...
1
vote
0answers
168 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
vote
1answer
63 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
152 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 ...
1
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2answers
89 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 ...
2
votes
1answer
1k 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
votes
1answer
882 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} ...
0
votes
1answer
62 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
128 views

Find an estimation (using polar coordinates)

Consider the function $$ f(x,y):=\lVert x\rVert^{1-n}\ln(\lVert x\rVert)(\arctan(\lVert x-y\rVert))^{-\alpha},~~0<\alpha<n,~~n>1,~~(x,y)\in\Omega\times\Omega,~~~\Omega\subset\mathbb{R}^n $$ ...
1
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2answers
103 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
27 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 ...
0
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1answer
30 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] = ...
1
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0answers
260 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?
4
votes
4answers
454 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
77 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
votes
1answer
83 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 ...
2
votes
1answer
56 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
votes
2answers
46 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 = ...
1
vote
2answers
32 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 ...
1
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0answers
62 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)= ...
3
votes
1answer
1k 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
40 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 ...
1
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0answers
370 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 ...
5
votes
1answer
75 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 $y=r\sin\theta$. ...
0
votes
1answer
5k 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
44 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: ...
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1answer
190 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 ...
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2answers
2k 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
58 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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vote
1answer
675 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
votes
1answer
991 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
190 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
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0answers
140 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
votes
1answer
69 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
44 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 $$u=\sqrt{\lambda_1}\cdot L.$$ ...
0
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0answers
125 views

Polar equation of perimeter of half ellipse

x = Cx + a * cos(ang); y = Cy + b * sin(ang); Cx, Cy are cords of center. ...
0
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1answer
149 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: $$ r^2=x^2+y^2$$ ...