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
195 views

Defining a spiral in polar coordinates

I'm trying to find a general form for a spiral that fits the following criteria: the inner radius is $N$, and for any point $q$ on the spiral, the arc length from the start of the spiral to $q$ is ...
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2answers
44 views

Find the image of a ring

I'm working on the following problem: Find the image of the ring defined by $4 \lt x^2 + y^2 \lt 16 $ under the mapping $$F(x,y) = \left(\frac{x}{x^2+y^2} , \frac{y}{x^2+y^2}\right)$$ It looks to ...
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1answer
142 views

Plotting an angle on a graph

So I know, my origin "(0,0)", my angle "theta" degrees, and the distance from the origin, "d" Now I think I can work this out with polar coordinates, but really have no idea how to go about it. My ...
2
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1answer
382 views

How do I define the limits of a double integral in polar coordinates over an annulus?

Evaluate the double integral by re-writing them in polar coordinates: $\displaystyle\iint\limits_{R}\frac{y^2}{x^2}\ dA$, where $R$ is part of the annulus (ring) $9\leq x^2+y^2\leq 25$ lying ...
2
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1answer
55 views

What's the name of each pseudo-rectangle in a spherical surface?

Consider the common surface of a spherical segment crossed with a spherical wedge. This produces a pseudo-rectangle in the sphere surface, and a perfect rectangle in a mercator projection. What's the ...
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1answer
260 views

How to integrate over polar coordinates

Evaluate the following double integral by rewriting it in polar coordinates: $\displaystyle\iint\limits_Dxy\,dA$, where $D$ is the disc with center at the origin and radius 5 I have very little ...
2
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1answer
408 views

Test for symmetry for polar graphs

From a calculus book I'm reading: "Unlike the graphs of an equation in $x$ and $y$, the graph of an equation $r=f(\theta)$ can be symmetric with respect to the polar axis, the line $\theta = \pi/2$, ...
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3answers
4k views

Square root of complex number in polar or rectangular form

I am trying to find how to simplify: $$\sqrt{\frac{A+jb}{C+jd}}$$ My calculator errors out, giving a math error, and I don't know how else to solve this.
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1answer
96 views

Need a hint on what's wrong - polar coordinates

I'm asked to solve the following $$ \int^2_0 \int^\sqrt{4-y²}_0 \sqrt{4-x^2-y^2} dxdy $$ I thought about using polar coordinates: (1) $0 \le x \le \sqrt{4-y^2}$ is the upper half of a circumference ...
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2answers
414 views

How to verify a conversion to spherical coordinates?

Is it possible to verify if a conversion of an integral in Cartesian coordinates to spherical coordinates was done correctly other than revising it looking for mistakes? I mean, is there some kind of ...
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2answers
101 views

How to find the number of intersection for $ \rho =\frac{\theta} {2\pi+1} $ and $\rho =\frac {1} {2-\cos\theta} $

How to Find the number of intersection for curve $ \rho =\frac{\theta} {2\pi+1} $ and curve $\rho =\frac {1} {2-\cos\theta} $ .
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2answers
448 views

Kepler's First Law in 3D

Kepler's First Law in 2D polar is $$ r = \frac{p}{1 + \varepsilon\cos(\nu)}. $$ How can this be written to consider ellipses in ...
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0answers
394 views

Polar Integration over intersection of two circles

Let $C_0$ denote a circle centered at $(0,0)$ with a radius of $r_0$ and let $C_1$ denote a circle of radius $r_1$ centered at a point $(x_1,0)$. Assume that we are given some function, $\phi(r)$ ...
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0answers
53 views

From cartesian to polar, on a 'wavy' sphere surface

For a hobby project I'm trying to transform a wavy halfsphere surface into smaller segments. For this I need to be able to go from cartesian coordinates to polar coordinates. One of the formulas for ...
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2answers
389 views

question about continuity: using polar coordinates

Given a function $f\colon\mathbb R^2\rightarrow \mathbb R$ I want to study continuity. So I know the $\varepsilon-\delta$ and sequence criterion. Now we had polar coordinates in lectures: set ...
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1answer
200 views

Converting from polar to Cartesian coordinates.

I'm looking at some notes that I was given for my Calculus II class on converting from Cartesian to polar coordinates. Now I understand how to solve for r and $\theta $ but I'm looking at how she ...
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1answer
85 views

Polar parametrization surface intersection

here is my problem: I need some help, i need the parametrization of the intersection of this two surfaces: $\ z^2= x^2+y^2 $ $\ (x-1)^2+y^2=1 $ Well, i can do it with cartesian equations $\ ...
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2answers
413 views

limits of Surface area of revolution in polar co-ordinates.

My Question is Find the area of the surface generated by revolving the right-hand loop of the lemniscate $\;r^2=\cos2\theta\;$ about the vertical line through the origin (y-axis). I know the formula ...
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1answer
73 views

Another polar integral bounds question.

A plane region $R$ is determined by the inequalities $y\ge0$, $y\ge-x$, $x^2+y^2\le3\sqrt{x^2+y^2}-3x$. Sketch the region and find it's area. I have foregone sketching the area and tried to use ...
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1answer
185 views

area between two polar curves

I am trying to find the area between the following two curves given by the following polar equations: $r=\sqrt{3}\cos\theta$ and $r=1+\sin\theta$. I did the following: First, I found the points of ...
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0answers
77 views

Obtaining the cardioid by mirroring the square root function in a line

In what line of the plane $C_{W}$ is the cardioid $$p= 2 (1 + \cos\theta)$$ mirrored, from the branch of the function $$w=\sqrt{Z}$$ which takes positive values in $X>0$ and $Y=0$. Seriously this ...
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1answer
60 views

Area of $\left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2$

I've used the modified polar coordinates: $x = 3r \cos \theta$, $y =5r \sin \theta$, which got me to $$r^2 \le 9 \cos^2 \theta + 25 \sin^2 \theta$$ What now?
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1answer
106 views

Find polar equation from 4 polar points

Given $4$ polar coordinates $(3, -\pi/6)$, $(1, \pi/3)$, $(3, 5\pi/6)$, $(-3, 4\pi/3)$, graph and find the polar equation. I know that the general polar equation is $r = ep / 1+- e \cos (\theta)$. ...
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3answers
116 views

Finding a length of arc, what's wrong?

Find: $$ \int \sqrt{x^{2}+y^{2}}dl$$ $$L: x^{2}+y^{2}= Rx$$ (at image $p' = -R\cdot \sin(\phi)$ )
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4answers
177 views

Find the maximum value of $r$ when $r=\cos\alpha \sin2\alpha$

Find the maximum value of $r$ when $$r=\cos\alpha \sin2\alpha$$ $$\frac{\rm dr}{\rm d\alpha}=(2\cos2\alpha )(\cos\alpha)-(\sin2\alpha)(\sin\alpha)=0 \tag {at maximum}$$ How do I now find alpha? ...
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3answers
747 views

Smooth Pac-Man Curve?

Idle curiosity and a basic understanding of the last example here led me to this polar curve: $$r(\theta) = \exp\left(10\frac{|2\theta|-1-||2\theta|-1|}{|2\theta|}\right)\qquad\theta\in(-\pi,\pi]$$ ...
4
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1answer
1k views

Heat equation in polar co-ordinates

I was studying the heat equation, when i saw a new variant of it. Here's the statement: "the edge $r=a$ of a circular plate is kept at temperature $f(\theta)$. The plate is insulted so that there is ...
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1answer
114 views

Sketch the polar graph $r=e^{-2\phi}$

How are you supposed to sketch this type of polar graph? Are you supposed to somehow relate this to $\cos\phi+i\sin\phi$ but can polar graphs even have an imaginary axis?! I am thinking that you ...
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3answers
3k views

Why is the formula for the area of a cardioid $ \int_a^b \frac{1}{2} r^2 d \theta$

I've seen this expression in many places :$\int_a^b \frac{1}{2} r^2 d \theta$ and was wondering if someone can explain where this came from? I've noticed that it's sometimes explained in conjunction ...
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2answers
145 views

How do you find the maximum value of $r$ in a polar function?

I have $\, r=\cos\alpha +\sin2\alpha,\quad 0\le\alpha\le\frac{\pi}{2}.$ Do you then find $\dfrac{dr}{d\alpha}$ and let that $=0$ ? I am after just a few set of instructions.
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1answer
698 views

Find the area of the shaded region between $r=e^{\theta/2}$ and $r=θ$ .

That's the picture of the shaded region I have to find the area of. I'm totally stuck on this problem mainly because these two curves don't intersect so I'm not sure how to find the bounds of ...
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0answers
194 views

Curl in cylindrical coordinates

I'm trying to figure out how to calculate curl ($\nabla \times \vec{V}^{\,}$) when the velocity vector is represented in cylindrical coordinates. The way I thought I would do it is by calculating: ...
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2answers
130 views

Dirac delta from polar coordinates to cartesian coordinates

I have: $$k_x = k \cos\theta\\k_y=k\sin\theta$$ I would like to rewrite in terms of $k_x$ and $k_y$: $$\exp(in\theta)\,\frac{\delta(k-\alpha)}{k}$$ I start from: ...
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2answers
157 views

$f(x,y)=\langle y- \cos y, x \sin y\rangle$

$f(x,y)=\langle y-\cos y,x\sin y\rangle$ $C$ is the circle $(x-3)^2 + (y+4)^2 = 4$ orientated clockwise. Relevant theorems: Green's theorem (this is under the Green's theorem section of our book). ...
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2answers
3k views

Dirac delta in polar coordinates

Given $$x=r\,\cos\theta\\y=r\,\sin\theta$$ and $$x'=r'\,\cos\theta'\\y'=r'\,\sin\theta'$$ how can I express $$\delta(x'-x)\delta(y'-y)$$ in terms of the polar coordinates? And the more general ...
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2answers
4k views

Plotting in the Complex Plane

I just wonder how do you plot a function on the complex plane? For example,$$f(z)=\left|\dfrac{1}{z}\right|$$ What is the difference plotting this function in the complex plane or real plane?
4
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1answer
77 views

Polar coordinations - problem with r and $\theta$

let's take a look on Archimedean spiral. the polar equation is $r = \theta$. click here to look. but $\tan (\theta) = y/x$ and $r = \sqrt{x^2+y^2}$, so $r = \theta \rightarrow \tan(\sqrt{x^2+y^2}) ...
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1answer
40 views

Determining the correct upper bound for an integral in polar coordinates

This seems super easy. But i am just a little bit stuck here. Haven't done much calculus recently. Can someone help me out real quick? Thank you in advance!
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2answers
870 views

Integration of radial functions?

Let $f(|x|)$ be a integrable radial function in $\mathbb{R}^n$ ($|\cdot|$ denotes the euclidean norm as in convention). The following identity is used to simplify computations ...
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1answer
454 views

How did theta become equal to 3pi/4 here?

How did theta become equal to 3π/4 in this particular example? Find a set of polar coordinates (r,θ) of the cartesian point (-4,4) such that -2π ≤ θ ≤ 2π and a. r > 0 and θ > 0 b. ...
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1answer
249 views

Inaccuracy in numerical calculation of arclength of part of an ellipse

I am trying to numerically calculate the arclength of part of an ellipse according to: $$ L = \int_0^{\phi_s}\sqrt{r^2+\left(\frac{dr}{d\phi}\right)^2} d\phi $$ where $r$ is defined as: $$ ...
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1answer
1k views

Finding area between two polar curves using double integrals

I have a homework question that is asking me to find the area that lies: Inside the curve $r=2+cos(2\theta)$ But outside the curve $r=2+sin(\theta)$ I think I'm supposed to be using a double ...
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3answers
552 views

Trying to understand the meaning of symmetry

The picture below is the solution to the following problem as presented in my book: Find the area of the region that lies inside both curves $$r = 8 + \cos \theta \\r = 8 − \cos θ$$ According to ...
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1answer
230 views

Moment of inertia of a circle

A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$ Thank you
4
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1answer
253 views

Mexican Hat wavelet in polar coordinates

I'm interested in wavelet framework for polar coordinates. In the paper of Hou&Qin (2012) was proposed a general method for definition of MH wavelets on a certain manifold. In short, first we ...
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2answers
394 views

Find Cartesian equation of $r=\theta$

I solved this problem, but I'm not sure my answer is correct as it seems very complex (compared to the polar equation). Did I make some mistake along the way or is it the right solution? $$r=\theta$$ ...
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2answers
6k views

Don't understand how to use jacobian for transformation of coordinates

Hello. I fail to understand why the Jacobian matrix is used to transform Cartesian coordinates to polar coordinates. If I'm not misunderstanding, it is assumed that the matrix ...
2
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1answer
4k views

Find a Cartesian equation of $r = 4\cos\theta$

I was able to figure the substitutions inside the equation, but I'm stuck with the equation's manipulation that will give me the solution. What would be my next step? $$r = 4\cos\theta$$ $$r^2 = ...
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2answers
297 views

Parametrization of a curve in polar coordinates

I'm trying to change this parametrics equations to polar coordinates $$ X(t) = 2\cos(t) - \sin(2t) \\ Y(t) = 2\sin(t) - \cos(2t) $$ What i tryed to do was raise the two equations squared, sum ...
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1answer
170 views

Line integral of $F = r \times k$ on hemisphere

Exam revision - Verify Stokes theorem directly by explicit calculation of the surface and line integrals for the hemisphere $r=c$, with $z \geq 0$, where $F = r \times k$ and $k$ is the unit vector ...