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
410 views

How to show the normal density integrates to 1?

How could you show that the normal density integrates to 1? $$ \int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi \sigma^2}} e^{-(x+\mu)^2 / \sigma^2} dx = 1 $$
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2answers
114 views

Plotting a polar curve

The question is, to generate a polar graph using a graphing utility, and to choose parameter interval so that the complete graph is generated. $$r=\cos\frac{\theta}{5}$$ To find such an interval, we ...
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2answers
75 views

Show that the parameterized curve is a periodic solution to the system of nonlinear equations

First I tried to convert the system to polar coordinates. This only made things worse (unless I made some idiotic mistake). Can I plug in the given ellipse (rectangular coordinates) into the ...
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0answers
15 views

Finding the intersection of 2 coordinates in spherical coordinate system

Sorry in advance for messing up any math term or being confusing. I have the following data: lat1, lon1, alt1, v1, h1 and ...
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1answer
26 views

why is theta in a restricted interval? (Polar coordinates)

What is the polar equation of the circle of radius 1 whose centre lies at the cartesian point (1,0)? So I got the correct answer of r=2cos(θ) But then is says theta is in the interval (-π/2)≤θ≤(π/2) ...
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3answers
33 views

Is there any way to express $\theta=c$ as some function of $r$?

I recently found this: Desmos Graphing calculator. I tried to plot the equation $\theta=45$ but it gave me an error: Sorry, you can't graph $\theta$ as a function of anything yet. So I started ...
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3answers
144 views

Is Adobe Acrobat's icon a special function?

It looks like a function in polar coordinates. Is it a special function ?
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4answers
164 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
46 views

Intersection of circle and ellipse

I'm looking for the points of intersection of a circle $x^2 + y^2 = r^2$ ($r$ is known, origin is $(0,0)$) and an ellipse $(x - x_0)^2 / a^2 + (y-y_0)^2 / b^2 = 1$ ($a,b,x_0,y_0$ are known). ...
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0answers
10 views

Solid angle subtended in latitude-longitude maps

I need to scale a latitude-longitude map with the solid-angle each "pixel" subtend. How can I obtain the said solid angle starting from the $\phi$ and $\theta$ angles? Thank you very much
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2answers
28 views

Evaluating a polar double integral on the semi disc

The integral: $$\iint_D (x^2-y^2)\,dx\,dy$$ where $D$ is defined as: $$\{(x,y)\in \mathbb R^2 \mid x^2+y^2\le 1, x\ge 0\}$$ Context I have solved double integrals on quarter discs but this semi ...
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1answer
32 views

Area of a Self-intersecting Curve

I was doing some work finding the areas of rose curves. The rose curve is a polar curve given by the equation $$ r(\theta) = \cos{k\theta} $$ When $k$ is even, the area is $\pi/2$, and when $k$ is ...
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0answers
14 views

Radians : negative and positive values

Recently I have been reading books on DSP where I came across Polar co-ordinates. I understand that on Polar graph (4 quadrants) we have 0,pi/2,pi,3/2pi and 2pi radians as we move from one quadrant to ...
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0answers
63 views

My orbiting body is orbiting about the wrong focus of it's elliptical orbit… why? [closed]

I am coding in c++ and am computing the position of an orbiting body as a function of time. Everything is almost working. I have a nice elliptical orbit. Except, my orbiting body speeds up as it ...
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2answers
40 views

Finding Multivariable Limits

Is there any good way to find a multivariable limit other than switching to polar coordinates? For example, students each year are inundated with problems like $$\lim_{(x,y)\to ...
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1answer
81 views

Help with polar coordinates for physics problem

I need to solve a physics problem but don't know about polar coordinates properly, can anybody help with it? Suppose a curve which is a current carrying wire: $$r=\frac ...
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2answers
47 views

Angle of intersection of two polar curves

I'm trying to find the angle of intersection between two polar curves: $$\begin{cases}r= 5 + 3 \sin\theta \\ r' = 3\end{cases}$$ I've set them as $5 + 3\sin\theta = 3$ and got to $\sin\theta = ...
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1answer
35 views

Find finite area between curves [closed]

Find the finite area enclosed between $r= a \sin 4(\theta)$ and $r= a \sin 2(\theta)$ in polar coordinate system.
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3answers
29 views

Transforming a polar equation into a Cartesian equation?

Transform the polar equation to a Cartesian (rectangular) equation: $$r= \frac5{5cosθ + 6sinθ}$$ These equations really stump me, so if you could be more "heavy-handed" with the explanation, I'd ...
3
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1answer
59 views

Meaning of Rays in Polar Plot of Prime Numbers

I recently began experimenting with gnuplot and I quickly made an interesting discovery. I plotted all of the prime numbers beneath 1 million in polar coordinates such that for every prime $p$, ...
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2answers
35 views

Complex numbers in polar form

If we have two complex numbers, in polar form, as the numerator and denominator of a fraction, and we are asked to write them as a single complex number, is there an easier way to deal with them ...
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1answer
283 views

Find the area of the region that lies inside both curves $r = 5 \sin (2\theta)$, $r = 5 \sin (\theta)$

A friend of mine and I have this problem for homework, and he's my math tutor for all intents/purposes. He's spent a solid hour trying to figure this out, watching videos and testing different ...
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3answers
288 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
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1answer
62 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
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2answers
95 views

Polar form of a complex number

Question: Write the polar form of $$\frac{(1+i)^{13}}{(1-i)^7}$$ Well its obviously impractical to expand it and try and solve it. Multiplying the denominator by $(1+i)^7$ will simplify the ...
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1answer
43 views

Keeping the arc length constant between points in a spiral

I'm making a visualization of points in a logarithmic spiral and want to keep the arc length between points (image particles) constant. I read that in an Archemedian spiral arc length is ...
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2answers
21 views

Explanation of how to go from polar to parametric equations.

I was wondering how you can make a polar equation parametric, and I just don't get it. My book says that for $r = f(\theta)$, $x = f(t) \cos t$ and $y = f(t) \sin t$, but this makes absolutely no ...
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3answers
4k views

Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
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1answer
26 views

Logarithmic spiral appears inverted

I'm learning to code the equation for a logarithmic spiral for a graphics visualization. However, it appears to be inverted with the radius getting smaller (rather than larger) toward the outside of ...
3
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1answer
47 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
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1answer
84 views

Changing a double integral into polar coordinates

I have the double integral $\int^a_0\int^{\sqrt{a^2-x^2}}_0 e^{-(x^2+y^2+a^2)} dydx$ And I am asked to evaluate this by changing to polar coordinates. I know the transformations are, $x=r ...
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0answers
37 views

What distinguishes elliptical coordinates from polar coordinates?

I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions. Polar: $$ x=r \cos(t) \\ y=r \sin(t) $$ ...
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1answer
25 views

Equation Conversion: Polar to Rectangular

Convert the polar equation to rectangular form (rectangular equation) $$r=\frac{9}{1-3\cos(\theta)}$$ I know that $r^2= x^2+y^2, x= r\cos(\theta)$ and $y= r\sin(\theta)$ and $\tan(\theta)= ...
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1answer
47 views

How to prove that the graph of $r=\sin(\frac{\theta}{2})$ is symmetry about polar axis

I want to know how to prove that the graph of $r=\sin(\frac{\theta}{2})$ is symmetry about the $x$-axis(polar axis). As I understand, if a polar graph is symmetrical about $x$-axis, $(r,\theta)$ and ...
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1answer
38 views

Harmonic functions and polar differential forms

Given a harmonic function $u$, its differential and conjugate differential are $$du = \frac{\partial u}{\partial x}dx + \frac{\partial u}{\partial y}dy,\qquad ^{*}du = -\frac{\partial u}{\partial y}dx ...
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2answers
54 views

Finding the length of a spiral

I need to find the length of a spiral. The spiral start at a certain radius $R_1$ and ends at a smaller radius $R_2$. As the spiral spins inwards, the distance between each arm of the spiral decreases ...
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2answers
30 views

Find the area of the circle

Find the area of the circle defined by the parametric equations $x = \cos t$ and $y = \sin t$. I know this is circle defined by $x^2 +y^2 =1$ so i used $0 < t < 2\pi$ as my bounds, then ...
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1answer
25 views

Compute double integral on polar coordinates, find $r(\phi)$

I have the function $f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$ and the region $\{y\geq 2x^2-2, y\leq 3x\}$. The region is: To compute the integral in cartesian coordinates: ...
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0answers
17 views

How many kinds of simple coordinates are there in a 2D space?

The question comes form an idea to solve a motion-with-potential problem in 1D space by finding a mathematically equivalent uniform-motion problem in 2D space. ...
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2answers
35 views

Converting from set of Cartesian equations to Polar Equation

Is it possible to convert the set of Cartesian equations: $$x(t) = (20-30)*\cos(2t)+45*\cos(2t*(20-30)/20))$$ $$y(t) = (20-30)*\sin(2t)+45*\sin(2t*(20-30)/20))$$ where $$t \in [0,2\pi)$$ Into a ...
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2answers
64 views

Integrating $ \int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} dy\,dx$ in polar coordinates

I'm having a problem integrating $ \displaystyle\int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} \,dy\,dx$. I drew the graph, and it looks like half a circle on top of the $x$ axis. I tried ...
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3answers
56 views

Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates

I'm having a problem converting $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates. I drew the graph using my calculator, which looked like half a circle on the x ...
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3answers
45 views

graph the curve and find its length, $r=\cos^2(\frac {\theta}{2}) $

graph the curve and find it's length, $r=\cos^2(\frac {\theta}{2}) $ I graphed it and found that it was a cardioid (or a sideways heart). I am getting stuck on the arc length. this is what I have: ...
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1answer
73 views

find the area of the region that lies inside both curves $r=3+2\cos\theta; r=3+2\sin\theta$

find the area of the region that lies inside both curves $r=3+2\cos\theta ; r=3+2\sin\theta$ The points of intersection should be $\frac {\pi}{4} and \frac {5\pi}{4} $ I don't think these graphs are ...
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1answer
14 views

Finding self-intersections on a polar curve

I have a polar curve $r = \frac{2}{\theta}$ (which is a hyperbolic spiral) and I need to find out where it self-intersects. When $\theta$ is restricted to positive values it never intersects, but when ...
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2answers
30 views

Is equation for ellipse in polar coordinates correct?

Wikipedia gives the following equation for the conic sections in the polar coordinate system: $r = \frac{l}{1+e\cos\varphi}$. According to the article on conic sections, in case of an ellipse $e = ...
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1answer
31 views

Find points near end point of a line

Any equation to find points near to both start and end points of lines with different slopes. See image. Need P and Q. If Endpoints are named A and B, AP and BQ should be 1 cm
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0answers
12 views

Heat equation on a circular plate

I'm in trouble with the following problem: assuming a circular plate of radius $R$, the heat equation on it, is: $$\partial_t ...
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1answer
86 views

graph polar coordinates $ r=4\sin(3\theta) $

Graph polar coordinates $ r=4\sin(3\theta) $ I was told by my teacher to split the graph into $3$ parts per quadrant and try those angles the problem arises when I plug $\dfrac{5\pi}{6}$ into the ...
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1answer
29 views

Tranforming to polar co-ordinates

$$I = \int_0^1\int_0^{\sqrt{1-x^2}} xy \, dy\, dx$$ By transforming to circular polar co-ordinates, evaluate I. How do I do this? Is there a formula/strategy for doing this that works with ...