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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votes
2answers
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+50
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
...
0
votes
1answer
18 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!
1
vote
1answer
45 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. ...
4
votes
1answer
105 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 ...
1
vote
1answer
44 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:
$$ ...
1
vote
1answer
34 views
Finding area between two polar curves using double integrals
I have a homework question that is asking me to find the area that lies:
Inside the curve $r=2+cos(2\theta)$
But outside the curve $r=2+sin(\theta)$
I think I'm supposed to be using a double ...
3
votes
3answers
63 views
Trying to understand the meaning of symmetry
The picture below is the solution to the following problem as presented in my book:
Find the area of the region that lies inside both curves $$r = 8 + \cos \theta \\r = 8 − \cos θ$$
According to ...
-1
votes
1answer
56 views
Moment of inertia of a circle
A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$
Thank you
0
votes
2answers
355 views
Single variable integral to polar coordinates?
I took calculus about 2 semester ago, and I'm trying to brush up on polar coordinates.
I integrated $-x^2+3$ from $x = -\sqrt{3}$ to $\sqrt{3}$ and I got $6.93$
Now I tried to convert it to polar ...
2
votes
2answers
87 views
Find Cartesian equation of $r=\theta$
I solved this problem, but I'm not sure my answer is correct as it seems very complex (compared to the polar equation). Did I make some mistake along the way or is it the right solution?
$$r=\theta$$
...
0
votes
2answers
45 views
Don't understand how to use jacobian for transformation of coordinates
Hello. I fail to understand why the Jacobian matrix is used to transform Cartesian coordinates to polar coordinates.
If I'm not misunderstanding, it is assumed that the matrix ...
2
votes
1answer
56 views
Find a Cartesian equation of $r = 4\cos\theta$
I was able to figure the substitutions inside the equation, but I'm stuck with the equation's manipulation that will give me the solution. What would be my next step?
$$r = 4\cos\theta$$
$$r^2 = ...
3
votes
0answers
53 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 ...
1
vote
1answer
31 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 ...
1
vote
1answer
108 views
triple integral - ecliptic coordinate
I need to find the $V$ by triple integral.
the limits from up is (1) - ecliptic cone.
and from dwon - (2) - elepsoide.
$$(1) \ \ \ \ z=-\sqrt{3x^2+5y^2}$$
$$(2) \ \ \ \ {3 \over 10}x^2+5y^2+{z^2 ...
0
votes
1answer
73 views
How does one interpolate between polar coordinates?
I'm finding that when I try to use the standard methods of interpolation in polar space, the result is not what I would expect. For example, when interpolating between the following polar coordinates:
...
1
vote
1answer
60 views
Polar Coordinates: Dividing by the variable “r.”
Evaluate the iterated integral by converting to polar coordinates:
$\large \int^2_0 \int^{\sqrt{2x-x^2}}_0 xy~dy~dx$
I successfully completed most of the problem; however, I had difficulty ...
1
vote
1answer
1k views
Find the equation in polar coordinate form for a straight line through the points with polar coordinates (4,0) and (4,π/3).
Find the equation in polar coordinate form for a straight line through the points with polar coordinates $(4,0)$ and $(4,π/3)$.
Here's my steps:
1.Write the two points in cartesian coordinates: the ...
0
votes
0answers
43 views
Which complex number cannot be written in polar form?
I'm really confused by this question. Is there such a number?
0
votes
1answer
458 views
Find the area of the Rose's petal.
If a Rose leaf is described by the equation $r = \sin 3\theta$, find the area of one petal.
0
votes
2answers
35 views
Finding the centroid of a polar curve
The curve is $r = e^{-b\theta}$ where $b > 0$ and $θ \in [0, \infty)$.
I got that the arc length is $\frac{\sqrt{b^2 + 1}}{b}$ (is this correct?), but computing the centroid $(x, y)$ looks awful. ...
2
votes
1answer
470 views
Polar Coordinates and Double Integrals
Problem 1:
Find the area enclosed by the ellipse $\displaystyle \frac {1} {r} = 1 – 0.6 \cos(\theta)$.
We know $0\leq \theta\leq 2\pi$.
We know $0\leq r\leq 1/(1-0.6\cos(\theta))$.
Questions:
...
0
votes
1answer
245 views
How to find the shortest distance from a line given in polar coordinates and a point given in Cartesian coordinates?
How do I find the shortest distance from a line given in polar coordinates and a point given in Cartesian coordinates?
For example, say that the line is given by the polar coordinates rho = 2 and ...
3
votes
1answer
164 views
How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?
How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$?
I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...
5
votes
6answers
735 views
Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?
Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
3
votes
1answer
216 views
Computing gradient in cylindrical polar coordinates using metric?
I am trying to understand coordinate transformations properly (having
studied some general relativity in the past).
Let us consider the transformation from cartesian to cylindrical coordinates,
...
1
vote
2answers
78 views
Coordinate system conversion: what it is called what I'm doing?
I want to do a simple coordinate transformation and would like to know what is the rigorous way to describe this mathematically in order to be able to search for algorithms for more complex ...
1
vote
2answers
64 views
Integration, polar coordinates
My question is general rather than specific.If a problem requires to find the area of a figure bounded by a curve given in polar coordinates,how do we find the limits of integration analytically ...
3
votes
1answer
37 views
polar coordinates ..question about the answer from the solution manual
Im trying to figure out but for some reason I dont know how to...could someone please tell me how did they get this answer from the solution manual....they skipped steps so I have no idea
1
vote
1answer
43 views
Express in Rectangular Form
a) $(-1+i)^{-i}$
so I know that the answer is $9.92-3.58i$. My track getting there is off.
I know that $x=-1$ and $y=1$, so $r = \sqrt{2}$, also that $\displaystyle \theta=-\frac{pi}{4}$.
I've ...
2
votes
0answers
30 views
Pure differential equation whose solution is a siluroid?
I am trying to find a differential equation for the siluroid that DOES NOT contain explicitly $\theta$, $\sin\theta$, or $\cos\theta$, but only $\rho$, $\dot\rho$, $\ddot\rho$. The siluroid equation ...
0
votes
0answers
32 views
Polar fourier transform
I need help please.
I have a 2D signal : sg=sin(x+y). to represent it in 2D i use meshgrid: [xx,yy]=meshgrid(x,y) and i represented with surf(xx,yy,sg).
Now i want to transform my signal in polar ...
2
votes
4answers
71 views
Converting x^2 + 6y - 9 = 0 to polar
Hi I'm trying to solve this problem but am having difficulty removing the remaining r. I have tried http://i.imgur.com/iJk9b2g.jpg but cannot get an answer
Any help is appreciated
1
vote
1answer
46 views
Polar coordinate
Let $f(x,y)$ be a differntiable function in $\mathbb{R}^2$ so that
$f_x(x,y)y=f_y(x,y)x$ for all $(x,y)\in\mathbb{R}^2$.
Find $g(r)$ so that $g(\sqrt{x^2+y^2})=f(x,y)$ and $g$ is differentiable in ...
2
votes
1answer
53 views
evaluation of double order integral using polar co-ordinates
When evaluating double integral using polar co-ordinates,
does the order of $dr ~ d\theta$ make any difference?
Suppose,
$$\int_0^{\pi/4}\int_0^{\sin\theta} r^2 dr d\theta$$
...
1
vote
1answer
60 views
How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation
Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 ) $. Evaluate the integral :
$$
\iint_D e^{\frac{y-x}{y+x}}
$$
a) by transforming to polar coordinates
b) by using the ...
1
vote
0answers
46 views
gradient of an axis symmetric vector field in cylindical coordiantes
I am trying to calculate with a general approach the gradient of an axis symmetric vector field in cylindrical coordinates and then express it in cartesian coordinates.
First I write my vector ...
1
vote
1answer
121 views
map on a unit sphere with polar coordinates
My brother, who is in hospital atm and cannot verify by himself asked me to post the following question, thank you in advance, and sorry if the topic has already been covered, i do not have the math ...
0
votes
2answers
19 views
Polar coordinates that uses $\frac { 1 }{ Z_1 }$
I am doing polar coordinates, and I am stuck when my book asks to do $\frac { 1 }{ Z_1 }$. I have no problems with $\frac { Z_1 }{ Z_2 }$ and $Z_1Z_2$. Here is the values for $Z_1$ I'm not so much ...
1
vote
2answers
143 views
How do I calculate numerically a tensor in polar coordinates?
You can formulate the question also like this: What is the easiest way of calculating directed derivative of a function if its values are evaluated in a cartesian grid?
a) fit a (spline) surface, ...
1
vote
1answer
58 views
What happens to a line in polar coordinates when orgin is moved and rotated in cartesian coordinates?
Let's say we have an Archimedean spiral in Cartesian coordinates. This corresponds to a line in polar system (i.e. $r=a\theta+b$).
Now if I move the origin of the Cartesian coordinates system to ...
0
votes
1answer
49 views
Polar form $\frac{dy}{dx}$
Trying to find the derivative $\dfrac{dy}{dx}$ in polar form, where:
$$x=r\cos\theta \,\text{ and } \, y=r\sin\theta$$
Seems like the common approach (on Wikipedia and other sites) is to assume that ...
0
votes
2answers
87 views
What is the inverse $z^{-1}(z)$ of $z(\varphi)=e^{i\varphi}$ with $\varphi\in\Bbb N_0$?
Suppose I am given a complex number $z=x+iy\in\Bbb C$, with $\left|z\right|=1$, and I am told that $z=e^{i\varphi}$ for some integer $\varphi\in\Bbb N_0$ (the value of which is not given to me).
How ...
2
votes
1answer
112 views
Really Stuck on Partial derivatives question
Ok so im really stuck on a question. It goes:
Consider $$u(x,y) = xy \frac {x^2-y^2}{x^2+y^2} $$ for $(x,y)$ $ \neq $ $(0,0)$ and $u(0,0) = 0$.
calculate $\frac{\partial u} {\partial x} (x,y)$ and ...
1
vote
1answer
900 views
Ellipse in polar coordinates
I think Wikipedia's polar coordinate elliptical equation isn't correct. Here is my explanation: Imagine constants $a$ and $b$ in this format -
Where $2a$ is the total height of the ellipse and $2b$ ...
0
votes
0answers
66 views
How to solve following non-linear differential equation?
Let's have an equation
$$
\left(\frac{\partial f}{\partial r}\right)^{2} + \frac{1}{r^{2}}\left(\frac{\partial f}{\partial \varphi}\right)^{2} = g(r).
$$
How to solve it?
0
votes
1answer
31 views
Qualitative analysis of an ordinary differential equation in polar coordinates
I want to draw the integral curves of the differential equation in polar coordinates $(\theta, \rho)$
$\frac{d\rho}{d\theta}= \rho^3-6\rho^2+8\rho$
At first I thought it would suffice to analyse ...
1
vote
1answer
41 views
Polar coordinates parameters
Sketch in the same diagram the curves with polar equations $r=2a\cos\theta$ and $2r(1+\cos\theta)=3a$ and find the polar coordinates of their points of intersection. What is the polar equation of ...
1
vote
3answers
111 views
Converting $x^2 + 6y - 9 = 0$ to polar.
So far I got here
\begin{align}
(r\cos\phi)^2 & + 6 r \sin\phi- 9 = 0\\
(r\cos\phi)^2 & = 9 - 6r \sin\phi
\end{align}
0
votes
0answers
71 views
How to remember symmetry tests for polar graphs?
Polar $(r , -\theta)$ & $(-r, \pi - \theta)$
Pole $(-r, \theta)$ & $(r, \pi + \theta)$
$\frac{\pi}{2} (-r, \theta)$ & $(r, \pi - \theta)$
