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 ($\theta$).
2
votes
1answer
2k views
Transforming the Laplace operator from Polar to Cartesian coordinates
I'm trying to find the error in my logic here.
Let's say we are given the Laplace operator in polar coordinates:
$$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...
1
vote
0answers
470 views
Conversion of motion equation from Cartesian to Polar coordinates: Is covariant differentiation necessary?
Say I have the following equation of motion in the Cartesian coordinate system for a typical mass spring damper system:
$$M \; \ddot{x} + C \; \dot{x} + K \; x = ...
2
votes
2answers
141 views
How do I find the limit of $\frac{xy\sqrt{|xy|}}{x^2 + xy + y^2}$ as x and y approach zero?
I am trying to find: $$\lim_{(x,y)\to (0,0)}\frac{xy\sqrt{|xy|}}{x^2 + xy + y^2}$$
I suspect that the limit does exist as the combined power of $x$ and $y$ is higher in the numerator than in the ...
4
votes
1answer
375 views
Polar coordinates, line integrals, and the Beltrami Identity
Imagine you are walking along the xy-plane. There is a landmark at the origin of the plane which distorts time at every point on the plane, such that the distortion is a function of the distance ...
0
votes
3answers
171 views
Visualizing why a right-angle rotation formula works in polar coordinates
I am trying to get a solid and intuitive handle on polar and spherical coordinates, and I'm getting stuck with what I think should be simple geometry:
To find the unit vector in Cartesian coordinates ...
1
vote
1answer
58 views
Am I doing this double integral right?
I want to calculate $\iint_R x \ \mathrm{d}A$, where $R$ is the unit disc centered at $(2, 0)$.
First, I made the following substitution: $$x' = x-2$$ $$\mathrm{d}x' = \mathrm{d}x$$ $$ ...
3
votes
3answers
1k views
Writing a Polar Equation for the Graph of an Implicit Cartesian Equation
If $(x^2+y^2)^3=4x^2y^2,$ then $r=\sin 2\theta$ for some $\theta$.
Using $r^2=x^2+y^2, x=r\cos\theta,y=r\sin\theta$, it's easy to get $r^2=\sin^22\theta$.
But I don't know what to do next, since ...
1
vote
3answers
85 views
Converting a polar coord to the range $0\le\theta\le2\pi?$
I know that you can keep adding/subtracting numbers to a polar coord, but what if I want to be able to take a number and just convert it to its positive equivalent?
2
votes
3answers
745 views
Ηοw to find the area of this region
I have two functions $$r=2$$
$$r= 3+2sin\theta$$
and I want to find the area of the yellow region in the picture below.
The limits of the integral solving the equation must be ...
2
votes
1answer
841 views
Horizontal and vertical asymptotes of polar curve $r = \theta/(\pi - \theta) \, , \, \in[0,\pi]$
I as supposed to find the vertical and horizontal asymptotes to the polar curve
$$ r = \frac{\theta}{\pi - \theta} \quad \theta \in [0,\pi]$$
The usual method here is to multiply by $\cos$ and ...
0
votes
0answers
66 views
Set of all points which are a specified angle away from a given point on a sphere.
I have a sphere with a known point on the surface in polar coordinates. I'm looking to find the set of all points which are exactly some angle away from this point in polar form (this should describe ...
0
votes
1answer
887 views
Question about the limits of integration using polar coordinates
I haven't been able to find an answer to something I've been thinking about.
If you are taking the integral of a circle in polar coordinates you always use the limits for theta as $0$ to $2\pi$. ...
1
vote
1answer
113 views
Bijections of the plane
I recently had to deal with polar coordinates and thus wondered: "Polar coordinates" is just a special name for some bijection from $\mathbb{R}^2$ to $\mathbb{R}^2$ that can be very easily visualized ...
3
votes
2answers
923 views
how to get $dx\; dy=r\;dr\;d\theta$
In polar coordinate how we can get $dx\;dy=r\;dr\;d\theta$?
with these parameters:
$r=\sqrt{x^2+y^2}$
$x=r\cos\theta$
$y=r\sin\theta$
Tanks.
0
votes
1answer
276 views
definition of sinusoidal curve
I have question related with these two definition:
In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates
$$r^n = a^n \cos(n \theta)$$
where $a$ is ...
1
vote
1answer
146 views
Questions about Hyperbolic Isometries: The Standard Inversion
I have two questions regarding the inversion across the unit circle in the hyperbolic plane.
Recall that the hyperbolic plane is a metric space consisting of the open half-plane
$$\mathbb{H}^2 = ...
2
votes
2answers
2k 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 ...
2
votes
1answer
177 views
Equation for the sensitivity pattern of a bi-directional microphone?
Can anyone give me an equation that expresses the sensitivity pattern of a bi-directional microphone, as a function of azimuth and elevation angle? A bi-directional microphone pattern looks something ...
4
votes
2answers
349 views
How do I write the 2D Dirac delta in a manifestly rotationally invariant form?
Consider the following integral over a 2D plane,
$$\iint \mathrm{d}^2\mathbf{k}\ e^{i\mathbf{k}\cdot\mathbf{r}} = 4\pi^2\delta^2(\mathbf{r})$$
This is a Fourier transform of a distribution which is ...
0
votes
1answer
264 views
3D parametric equations with polar coordinates
I'm currently studying for my calc 2 midterm and came across this and it completely lost me. I'm not even completely sure where to begin with it. Any ideas?
Put $\langle x[r,t],y[r,t],z[r,t] \rangle ...
0
votes
0answers
115 views
How to use polar coordinate to represent a $1 \times 1$ square rotated $45^{\circ}$ and translate to $(7,4)$?
How to use polar coordinate to represent a $1 \times 1$ square rotated $45^\circ$ and translated to $(7,4)$? Does the $r(\theta)$ have discontinuous (such at jump from $+5$ to $-2$)?
Please help. ...
0
votes
1answer
159 views
How to determine a shape is convex by giving polar form polynomial equation?
It is easy to determine concave, convex curve in xy coordinate. But I am placing a question that I only have a polar polynomial equation like r(ang) = a4*ang^4 + a3*ang^3 + .... + a0;
How I can tell ...
1
vote
2answers
394 views
Express this curve in the rectangular form
Express the curve $r = 9/(4+\sin \theta)$ in rectangular form.
And what is the rectangular form?
if I get the expression in rectangular form, how am I able to convert it back to polar ...
5
votes
6answers
746 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 ...
2
votes
2answers
95 views
Match the following polar equation to the best description: r^2 = 39 / sin(2θ)
Now, I've guessed the answer, it's a hyperbola, and I know what a hyperbolic function looks like, but I'm having a hard time getting it there. Here's my work so far:
First off, $r^2 = x^2 + y^2$
...
6
votes
2answers
3k views
Find the area of the region inside: $r= 6\sin(\theta)$ but outside of $r = 1$
How do we find the area of the region inside $r = 6 \sin(\theta)$, but outside $r = 1$?
So, here's my work thus far:
First off, we know:
$r^2 = x^2 + y^2$
and $\mathrm{sin}(\theta) = y/r$
...
1
vote
3answers
368 views
A square root of i with negative imaginary part
In an ODE class, one assignment question says find the “rectangular” expression z = a + bi (with a and b real) and the “polar” expression |z|, Arg(z) where z is "a square root of i with negative ...
4
votes
2answers
3k views
Why is $dy dx = r dr d \theta$ [duplicate]
Possible Duplicate:
Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$
I'm reading the proof of Gaussian integration. When we change to polar coordinates, why do we ...
3
votes
1answer
990 views
polar coordinates of Gaussian Distribution with non zero mean
I found that the polar coordinates of 2-dimensional Gaussian distribution with mean zero
$$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({x^2+y^2})/{2\sigma^2}\big) ...
1
vote
1answer
781 views
conversion of 2D Gaussian into polar coordinates
Is it possible to convert the 2D Gaussian function in to polar coordinates?
...
4
votes
1answer
253 views
Get polar equation from cartesian equation
I have this equation: $x^4 + y^4 = x^2 + y^2$ and I need to convert it to a polar one...
I have tried and the result is
$$r = \sqrt{\frac{1}{\cos^4\theta + \sin^4\theta}}$$
Is this ok?
3
votes
5answers
913 views
Length of $r=3\sin(\theta)$
I have a general understanding of calculating arc length, but this one's a real curve ball.
So, I need to find the exact length of $r=3\sin(θ)$ on $0 ≤ θ ≤ π/3$
So the way I've thought of ...
1
vote
2answers
218 views
Trying to plot these points in a polar coordinate system
I started with:
inside $r_1=5 \sin(θ)$ and outside $r_2=2+\sin(θ)$ and was told to sketch curve in the same polar coordinate system
I first set both equal to $0$ and solved to get $\pi$, $2\pi$, and ...
2
votes
2answers
394 views
Did I sketch this polar curve correctly?
The equation is:
$r^2=-4 \sin(2\theta)$
I first made a reference graph in cartesian coordinates using values $\displaystyle \frac{\pi}{4}$, $\displaystyle \frac{\pi}{2}$, $\displaystyle \frac{3 ...
1
vote
2answers
138 views
Sketching a polar curve
Continued off the question I asked earlier, I also have to sketch the curve.
$r^2=−4\sin(2\theta)$
So I have to set up a table of values I'm assuming. How do I know what values to choose for ...
1
vote
2answers
177 views
How to solve a polar equation when $r$ is $r^2$ instead?
I have $r^2=-4\sinθ$
and I'm asked to set $r=0$, then find θ. If I just set $r^2=0$ then I'll get $\sin(2θ)=0$. That doesn't seem right.
Then I'm asked to set $θ=0$ and then find $r$. If I use the ...
2
votes
1answer
480 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:
...
2
votes
1answer
222 views
Problem calculating an integral over a surface
I've been trying to solve this for awhile and can't find a way.
Given $ S={(x,y,z) \in R^3 : z = x^2 - y^2 , x^2 + y^2 \leq 1 } $ and $\phi :R^3 \to R $ defined as $\phi (x,y,z)= (4z +8y^2 + ...
6
votes
2answers
239 views
Need help with Curves and parameterizations
I'm having some trouble solving a couple of problems:
I know this one must be pretty easy but can't find the way to solve it.
I need to find the arc length of a curve described by $ r=1- ...
1
vote
1answer
231 views
Multivariable calculus double integral to polar coordinates
The task is to note down $\iint_D F(x,y)\mathrm dy\mathrm dx$ lane rows in polar coordinates. And region D is defined by $x^2 + y^2 = ax,\, a > 0 $ and $x^2 + y^2 = by,\, b > 0 $ intersection.
...
1
vote
0answers
50 views
Minimization of matrix of vectors in polar field
The problem I am facing is the reduction of vibrations of a rotating object. I have a series of vibration measurements taken at 5 different states with magnitude and phase components, and a set of ...
23
votes
4answers
2k views
Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$
It is changing the coordinate from one coordinate to another. There is an angle and radius on the right side. What is it? And why?
I got:
$2\mathrm dy\mathrm dx = r(\cos^2\alpha-\sin^2\alpha)\mathrm ...
0
votes
1answer
165 views
$\int_{0}^{6} \int_{0}^{y} x dx dy$ where $x = r \cos \theta, y = r \sin \theta, dx dy = r dr d \theta$
Given $x = r \cos \theta, y = r \sin \theta, dx dy = r dr d \theta$, how can I evaluate the following integral:
$\int_{0}^{6} \int_{0}^{y} x dx dy$
14
votes
4answers
594 views
What are the polar coordinates of the origin?
In polar coordinates, the origin has $r = 0$, but $\theta$ is not unique.
what sort of problems does this create, and how can I resolve them? For example, suppose an ant is wandering around a plane. ...
5
votes
2answers
215 views
Laplacian of a Function depending on r in Polar Coordinates
From a bank of exams:
Let $u(x,y) = f(r)$ be a smooth
function in the plane that depends
only on $r = \sqrt{x^2 + y^2}$.
Compute $\Delta u = u_{xx} + u_{yy}$
in terms of $f$ and its ...
1
vote
2answers
1k views
Add nautical miles to latitude and longitude decimal notation
What is the easiest way to add a set number of nautical miles to a known latitude and longitude? I am writing a program in C# that takes a point of origin in decimal notation:
33.4483333, ...
2
votes
0answers
177 views
What is the total area enclosed by a polar curve and the x-axis?
I am aware of the formula.. but can someone give me a clear definition of what the total area enclosed by a polar curve represents.
Thanks
1
vote
2answers
3k views
dA in polar coordinates?
I have seen a picture for $dV$ so that $dV = r^{2} \sin(\theta) dr d\theta d\phi$. But how can I deduce things like dA and dV? In a simpler coordinate (not sure about the name), $dA = r dr d\phi$, ...
0
votes
0answers
150 views
How do you create a spectral plot and then later a heat or intensity map from 2d data?
I'm a programmer, but I'm no mathematician, so I need a little help understanding what is required to accomplish this task.
Here's a short explanation of the data I have from the Coastal Data ...
1
vote
2answers
859 views
Converting polar equation to cartesian coordinate polar equation and back again?
OK, so I have the following polar equation:
$r = Θ/20$
And I would like to translate this a little to the right, and down from the polar origin.
Now, I figure since I know cartesian coordinate ...
