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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4
votes
1answer
331 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
2k 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
263 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
655 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
253 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
220 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
715 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
406 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
281 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
382 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
57 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 ...
27
votes
4answers
3k 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
221 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
1k 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. ...
6
votes
2answers
320 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
2k 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
202 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
2
votes
2answers
7k 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 ...
1
vote
2answers
1k 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 ...
6
votes
1answer
2k views

polar coordinates and derivatives

Using the standard notation $(x,y)$ for cartesian coordinates, and $(r, \theta)$ for polar coordinates, it is true that $$ x = r \cos \theta$$ and so we can infer that $$ \frac{\partial x}{\partial ...
5
votes
4answers
3k views

How to deduce the area of sphere in polar coordinates?

$A = r \int_{0}^{\pi}\int_{0}^{2\pi} e^{i (\alpha+\theta)} d\alpha d\theta = r \int_{0}^{\pi} [\frac{-i}{\alpha+\theta} e^{i(\alpha + \theta)}]_{0}^{2\pi} d\theta = ... ...
1
vote
2answers
115 views

What is a good way to pick points for polar equations?

If I want to plot $r = 4\sin3\theta$ what is a good way to pick points? The period is $2\pi/3$ so it will repeat after that, but how do I pick points so that I get a good range of values so that I ...
1
vote
1answer
143 views

optimum rectangle around a group of coordinates (longitude and latitude)

Okay, I am in no way a mathematics student but I happened to be writing a program on a map with coordinates of locations displayed on the map. My problem is that the points could be scattered on the ...
4
votes
2answers
1k views

Finding the volume of water in a swimming pool

Reposting from stackoverflow :) - Told to go here :) So I've got a issue with a math assignment, and I were hoping someone could at least point me to what I should be doing, because currently I'm ...
1
vote
2answers
780 views

Difficulty with differentiation in polar coordinates

I am trying to understand a simple calculation in polar coordinates - and I am getting totally discombobulated (the original source can be found: here).Please have a look at the following picture: ...