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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717 views

How to calculate the polar arc length of the entire cardioid $r=a(1-\cos\theta)$

I'm having a bit of an issue calculating the arc length of $r = a(1-\cos\theta)$. I'll begin by listing the steps I made in my attempt to solve this exercise. We know that the arc length formula ...
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1answer
38 views

What is the area of the closed curve?

The graph of the polar graph $r=\dfrac{4}{2-\cos\theta}$ forms a closed curve. Find the area of the region inside the curve.
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890 views

What is the graph of the polar equation theta = pi?

The question exactly goes like the title. I'm thinking that it's a point on the 3.14, but as I'm typing this I realize that I'm wrong and now I'm out of clues (Google didn't help). Please enlighten ...
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2answers
82 views

Use Polar Coordinates to Find the Limit…

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geq 0$, $r \to 0^+$ as $(x,y) \to (0,0)$)] $$\lim \limits_{(x,y) \to (0,0)} ...
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42 views

Evaluating an integral over inifinty with polars leads to an integral of cosine over inifinity, how can this be resolved?

So I have the integral $$\int_0^\infty\int_0^\infty\frac{yx^2}{x^2 +y^2}e^{-(x^2 +y^2)} \,dx\,dy$$ And converting this into polars gives: $$\int_0^\infty r^2 e^{-r^2}\,dr ...
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1answer
36 views

What am I missing converting cartesian to polar coordinate system?

I've got the equation $ x^2+y^2=2x $. By looking at the graph of that function, I know that it is equivalent to $ r=2\cos{\theta} $ (graph). However, if I convert it by substituting in using the ...
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1answer
50 views

Problem with Laplacian while treating polar coordinates as special case of spherical coordinates.

I thought that polar coordinates ($r, \phi$) can be viewed as a special case of cylindrical coordinates ($\rho, \phi, z$) with $z=0$, or as spherical coordinates ($r, \theta, \phi$) with ...
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2answers
155 views

Converting between polar and Cartesian coordinates

The polar coordinates $r$ and $\varphi$ can be converted to the Cartesian coordinates x and y by using the [[trigonometric function]]s sine and cosine: $$x = r \cos \varphi \,$$ $$y = r \sin \varphi ...
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363 views

Finding the area bounded by $r = a(1-\sin\theta)$ and $r = a$

Consider the cardioid $r = a(1-\sin\theta)$ and the circle $r = a$. We have that the cardioid meets the origin at an angle of $\frac{\pi}{2}$, while it reaches its maximum distance from the origin at ...
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225 views

Show that the four points given below are the vertices of a rhombus.

Show that the four points, $(5, 8), (7, 5), (3, 5)$ and $(5, 2)$ are the vertices of a rhombus. I tried solving it, by finding out the distances by using the formula $\sqrt{(x_{2}-x_{1})^2 + ...
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48 views

Finding intersections points of pairs of polar curves?

Find all intersections of the curves $r=3^{(1/3)}\cos(\theta) , r=\sin(\theta)$ What I have done so far is to just put them equal to each other like this: $3^{(1/3)}\cos(\theta)=\sin(\theta)$, but ...
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1answer
36 views

Sketch the polar graphs

Any advice on how to sketch polar graphs? I have tried transforming to rectangular coordinates, but its not really much help $$ r=1+\sin(\theta) \\ r^2=4\cos(2\theta) $$ Thanks in advance :)
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1answer
57 views

transforming cartesian to polar coordinates?

Transform the given polar equation to rectangular coordinates, and identify the curve represented. $$r=\frac{5}{3\sin\theta-4\cos\theta}$$ Any tips? The first thing I tried was replace $\sin\theta$ ...
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2answers
185 views

Mathematical roses with $4n+2$ petals

In polar coordinates $(r, \theta)$, the equation $$r = \sin\left(a \theta\right)$$ gives a rose with $a$ petals if $a$ is odd, or $2a$ petals if $a$ is even. Thus, the number of petals generated for ...
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49 views

Polar to phasor

Let's say that there is a polar equation: -2400 + 8320j To convert this polar equation to phasor form, should the negative be considered when trying to find the angle? Would the angle be +73.91 or ...
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1answer
50 views

Complex number to polar representation

I'm trying to change the complex number, $-3i$ to polar representation. What I did: $a=0$ $b=-3$ $r=\sqrt{a^2+b^2} = \sqrt{0+3^2} = 3$ $\theta = \frac{b}{a} = \frac{-3}{0}$ But after that I'm ...
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61 views

How do i justify integration by polar-coordinates for Riemann-integration?

I completely understand how to transform Lebesgue integration to integration by polar-coordinates using the surface measure. However, i wonder if there is a weaker version of this justifying ...
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26 views

Find the polar equation of a graph / graph a polar equation

What general approach can I take to find the polar equation of a given graph in the polar coordinate system? What general approach can I take to graphing any polar equation in the polar coordinate ...
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1answer
316 views

Definition of “the surface measure”?

Let $\mu_n$ be the $n$-dimensional Lebesgue measure. Let $||\cdot||$ be a norm on $\mathbb{R}^n$. Define $S^{n-1}=\{x\in\mathbb{R}:||x||=1\}$. I have proven that $\forall A\in\mathscr{B}_{S^{n-1}}, ...
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1answer
183 views

Find rectangular equation of a cardioid

Given the equation in polar form $$r = 1 - \sin\theta,$$ find the rectangular equation. So far, I found: $$x^2 + y^2 = 1 - 2\sin\theta + \sin^2\theta\quad x = \cos\theta - \sin\theta\cos\theta\quad ...
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1answer
38 views

How to find polar values of complex number as quick as possible?

I need to calculate these kind of values in exams in best speedy way. Convert $1.46 + 3.17j$ to polar form ($r∠θ$) Is there is any solution to find of the values as quick as possible? By the way, ...
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1answer
40 views

Reference for studying polar coordinate

There is a theorem about justification of polar-coordinate in Folland-Real analysis p.78. I find it somewhat terse (Maybe it's just me).. I guess this kind of transform is possible even when ...
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111 views

Cartesian vector field to vector field

Ok so I have a given vector field in Cartesian coordinates, say \begin{align*} \textbf{v}(x,y)=\frac{dx}{dt}\hat{\textbf{e}}_{1}+\frac{dy}{dt}\hat{\textbf{e}}_{2} \end{align*} Where $dx/dt$ and ...
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52 views

Rectangular transformation into Polar coordinates

I was working with a simple transformation of rectangular coordinates - symmetry around the y-axis, i.e. $$f(x,y) = (x, -y)$$ I wanted to express the identical concept in polar coordinates. After ...
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1answer
54 views

Polar equation and Cartesian equation

For the polar equation, $r \sin \theta = \ln r + \ln (\cos\theta)$ Is that equivalent to $y = \ln x $ ?
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1answer
52 views

For which $\alpha \in \mathbb{R}$ does $\int_{\mathbb{R}^n} \big(1+|x|\big)^{\!-\alpha} \mathrm{d}x$ exist?

I assume only $\alpha \gt 1$ gives $\int_{\mathbb{R}^n} (1+|x|)^{-\alpha} \mathrm{d}x \lt \infty$ (simply because this is true for $n=1$). I also assume some clever transformation could be used for ...
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119 views

Change to polar coordinates when evaluating limits of functions in two variables?

I have a function in two variables $f(x, y)$ and need to calculate the limit $$ \lim_{(x, y) \rightarrow (2, 3)}{f(x, y)} .$$ If I decide to change to polar coordinates, how can I determine where $r$ ...
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382 views

Sketch the polar graph r=2+cos(theta). Find the points of intersection, if any, of this graph with the straight line y=2x-1 (use two decimal places)

I have already sketch the polar graph. and I have to find this graph's intersection point with the straight line y=2x-1 so, I try to solve it like this way: y=2x-1 Rsin(theta)=2Rcos(theta)-1 ...
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85 views

Problem understanding solution of complex nth-root of unity

a while ago we had the solution for a complex number task about the nth-root of unity in the complex. But now I am having some difficulties to fully understand it: The task was to find all complex ...
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135 views

How to calculate $\theta$ when we know $\tan \theta$.

Hej I'm having difficulties calculating the angle given the tangent. Example: In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this ...
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83 views

Velocity of a particle in polar coordinates

The equations $r = 3\sin(2\theta)$ and $\frac{d\theta}{dt} = 2$ describe the motion of a particle in polar coordinates. Find the velocity of the particle in terms of the unit vectors $u_r$ and ...
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16 views

The Set of Closed Curves Representable by $r(\theta)$.

I apologize in advance if my terminology and/or notation is inaccurate; I am a little out of my depth here. If something is unclear, please point it out and I will try to explain myself better. ...
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43 views

Is it possible to change the pole and/or the polar axis in a polar coordinate system?

Citing Wikipedia's article on polar coordinates... In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance ...
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334 views

how to find existence and value of limit in multivariable calculus

I was in maths class and i found a question interesting. Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar ...
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1answer
59 views

Area of a sphere bounded by a paraboloid

I need to find the area of the surface $x^2+y^2+z^2 = a^2$ for $y^2 \ge a(a+x)$. I know that $A = 4a \int_{-a}^0 dx \int_{\sqrt{a^2+ax}}^{\sqrt{a^2-x^2}} \frac{dy}{\sqrt{a^2-x^2-y^2}}$, but I have ...
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1answer
34 views

Integrals in polar coordinates

Polar and spherical coordinates seem very useful for areas, however I don't understand why I can't seem to keep a direction after a spherical integral. In Cartesian coordinates, it's very easy to ...
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1answer
1k views

Area that lies inside both curves: $r=sin2\theta, r=cos2\theta$

My integral is setup as: $$A=8\int_0^\frac{\pi}8{\frac12sin^22\theta}\space d\theta - 8\int_{\frac{\pi}8}^0{\frac12cos^2 2\theta}\space d\theta$$ $$=8\int_0^\frac{\pi}8{\frac12sin^22\theta}\space ...
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2answers
450 views

Find the cartesian equation of: $r=2\cos\left(\frac {3\theta}{2}\right)$

I've managed to use identities to simplify it down to: $$r = 2\left(\cos^3\left({\theta\over2}\right)-3\sin\left({\theta\over2}\right)\cos\left({\theta\over2}\right)\right)$$ using trig identities, ...
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82 views

Hankel transform and Laplacian in cylindrical coordinates

My book solved a PDE containing the Laplacian in cylindrical coordinates. It doesn't really explain why the Hankel transform is useful in this case (symmetries etc..); just brute force math. So yeah, ...
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0answers
42 views

Derivatives of polar coordinates

I've got a problem for which I'm trying to calculate $\ddot r$. The problem is right here for the sake of reference. So far, I've got that: $$\ddot r=\frac{d}{dt}v_r=\frac{dv_r}{dr}\frac{dr}{dt}$$ ...
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1answer
96 views

Help with Polar coordinates and the length of the curve.

I have a test coming up today and I was going over our past midterms and this question came up. I tried it but its not working, please any hints or solution in how to do it will be really helpful. ...
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1answer
33 views

Polar Integral Confusion

Yet again, a friend of mine asked for help with a polar integral, we both got the same answer, the book again gave a different answer. Question Use a polar integral to find the area inside the ...
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1answer
31 views

Polar partial derivatives continuously differentiable implies holomorphic

I need to show that if $f(re^{i\vartheta}) = U(r,\vartheta) + iV(r, \vartheta)$ and $U,V$ are continuously differentiable and satisfy the Cauchy-Riemann equations, then $f$ is holomorphic. I am ...
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1answer
49 views

Can I solve for the fractional volume of a hyperboloid?

This looks like a homework problem because it is. I'm stuck at the portion where I solve for fractional volumes. Suppose you are a part of a team designing a water tank in the shape of a hyperboloid. ...
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1answer
66 views

polar coordinates, finding tangents

I've been asked to find the coordinates of the points on the curve: $r = acos2\theta, -\dfrac{\pi}{4} \leq \theta \leq \dfrac{\pi}{4} $ where the tangents are parallel to the initial line. The ...
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231 views

How to map sphere to faces of an Icosahedron

This is the mathematics behind some graphics I am trying to build in OpenGL. I believe the question belongs here. I want to represent an approximate sphere (let's say the Earth) as an icosahedron and ...
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1answer
51 views

Complex number in polar coordinates

I have to get $\Im$, $\Re$, the absolut value as well as the argument $\phi$ of the complex number $$z = \left(-\frac{1}{\sqrt2}+\sqrt\frac{3}{2}i\right)^8$$ I do this by transforming $z' = ...
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1answer
83 views

Polar coordinates integral

The exercise is Evaluate the double integral of the function $f(r, \phi) = r$ in the domain limited by cardioid $r = a(1 + \cos(\phi))$ and the circle $r = a$ If $T$ is the domain, I want $$\int ...
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3answers
2k views

How to convert a circle off origin to polar coordinates for integration?

I am trying to find the surface area of $x^2+y^2+z^2=a^2$ over the region $x^2 +y^2 \leq ax$. I rewrote the region as $\left(x-\frac{a}{2}\right)^2 + y^2 \leq \frac{a^2}{4}$. This is where I am stuck. ...
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87 views

Integration of figure whose base is a quarter circle not centered at origin using polar coordinates

How do I integrate $$ \int_{1}^{2}\int_0^{\sqrt{2x-x^{2}}}\frac{1}{\sqrt{x^2+y^2}}dydx $$ using polar coordinates? The base is a quarter circle of radius 1 centered at (1,0), so my first instinct ...