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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22
votes
3answers
2k views

How to force prime numbers into a line?

Inspired by an article on Prime Spiral and Hough transform I tried to analyze patterns created by plotting numbers on spiral (Archimedean?). $$x = \cos( angle ) * radius$$ $$y = \sin( angle ) * ...
0
votes
1answer
300 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
2
votes
1answer
17 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...
9
votes
3answers
204 views

Is Adobe Acrobat's icon a special function?

It looks like a function in polar coordinates. Is it a special function ?
0
votes
1answer
104 views

Help determining angle

Let $R$ be the triangle defined by $−x\tan(\theta) \le y \le x\tan(\theta)$ and $x \le 1$ where theta is an acute angle. Sketch the triangle and calculate \begin{equation*} \iint_R(x^2+y^2)\mathrm ...
0
votes
1answer
25 views

polar coordinates question

I was tasked with writing $\iint_D f(x,y) \,dx \,dy$ for $ [ D:{4\leq x^2 + y^2 \leq}9]$ through ''reoccurring integrals'' in polar and Cartesian systems? what are ''reoccurring integrals''? and how ...
1
vote
0answers
27 views

polar system in a plane?

what is a polar system in a plane and how it helps in calculating integrals in certain areas? I'm looking for a good explanation/a fair/ readable source on the matter.
2
votes
2answers
56 views

Find a solution that satisfies Laplace's equation in polar coordinates

How may I find a solution that solves Laplace's equation in polar coordinates, subject to the boundary conditions? In particular, I need to find one solution that satisfies $$\Delta u = 0,$$ subject ...
0
votes
0answers
14 views

Converting a polar equation in all variables to find properties of the corresponding linear equation.

Given the equation $\sin R = nx^q$, find the slope, $m$ and $y$-intercept, $c$, that corresponds to the linear form ($y=mx+b$) of the same equation. I understand this is a trivial question I am ...
0
votes
1answer
18 views

Geometric interpretation of $\frac{dr}{d\theta}$ in Cartesian Coordinates

You'll have to excuse me if this questions is extremely trivial; it's been years since I went back to elementary calculus and I humbly accept that I haven't really gotten deep into polar ...
1
vote
2answers
60 views

Solving non-linear second order differential equation: radius of curvature $= k \theta$

I'm trying to find any curve where the radius of curvature increases linearly with angular displacement. So in polar coordinates radius of curvature $= k \theta$ $$ \frac{(r^2 + r'^2)^{3/2}}{r^2 + ...
1
vote
1answer
38 views

Surface area generated by revolving $r = \sqrt {\cos 2\theta}$

I've been giving a good time trying to solve this problem, I do not find a clear way to solve appreciate your help. \begin{array}{rcl} r& =& \sqrt{\cos 2\theta } \end{array} This Around to ...
1
vote
1answer
24 views

Derivative of a polar coordinate equation

I was trying to plot the polar curve: $r=\cos(2n\theta)$ ($0\leq\theta\leq 2\pi$) and tried differentiating with respect to $\theta$ to get some information about where the petals would be. My ...
2
votes
1answer
36 views

Intersection of polar curve with line

I apologize for the horrible title. I came across this in an exam question: You're given $C_1$ as $$r = 1 + cos 2\theta $$ For $\frac{\pi}{2} \leq \theta \leq -\frac{\pi}{2}$. The only symmetry $C_1$ ...
0
votes
2answers
300 views

converting kph and heading to xyz velocity vector

I am writing software (in C++) that is required to send out messages from our simulation system to another simulation system. Problem is we track the simulation object's current speed (kph) and ...
-1
votes
0answers
24 views

Why is polar conversion not helpful in finding dominant direction in any image? [closed]

I tried to do polar conversion in finding dominant direction in any image. Although it is not helpful. What are the techniques that I could implement in Java to find the dominant direction of any ...
0
votes
0answers
123 views

Help me out with my assignment question? Area of triangle using double integral in terms of polar coordinates.

Let $R$ be the triangle defined by $-x \tan t \leq y \leq x \tan t$ and $x \leq 1$ where $t$ is an acute angle. Sketch the triangle and calculate double integral $(x^2 + y^2) dA$ using polar ...
2
votes
2answers
30 views

How to convert from cartesian to polar equation

I am trying to convert the equation $y=4/x$ into a polar equation. I have done this work but I am not sure if it is right. I just subsituted $r\sin(\theta)$ for $y$ and $r\cos(\theta)$ for $x$ and ...
2
votes
2answers
60 views

Find exact length of polar curve $ r= \frac{6}{1 + \cos \theta}$

I find myself frustrated with the solution of this problem since profit not find it, I'm stuck in the middle of the problem I can not solve the integral, I'm stuck in the solution of the integral is ...
0
votes
1answer
23 views

Area shape calculating

Can't find the area of the figure bounded by the curve in polar coordinates $$\phi=r\arctan(r), \phi=0, \phi=\frac{\pi}{\sqrt 3}.$$ I tried use the formula $$S=\frac 12\int_{0}^{\frac{\pi}{\sqrt ...
0
votes
1answer
19 views

How to prove this ODE is stable but not asymptotically stable?

Consider the ODE in polar coordinates: $$ r'=f(r),\theta ' =1 $$ where $$ f(r)=r\sin (1/r^2), r\neq 0, f(0)=0. $$ show that the origin is stable but not asymptotically stable.
2
votes
2answers
29 views

Finding the bounds of a solid for triple integrals

Ok, so I have an answer, most likely the wrong one. The question being asked is: Using polar coordinates find the volume of the solid bounded below by the $xy–plane$ and above by the surface $x^2 ...
0
votes
1answer
25 views

Finding limits of integration for double integral

Given a region where the $x$ limits are $-1< x<1$ and $0< y<\sqrt{4-x^2}$, with the option of converting into polar coordinates, i.e. the function $(x,y)$ can be replaced by $r^2$. I'm ...
0
votes
0answers
20 views

Normal form of plane

I propose a simple equation of plane generalized from 2D Normal form of straight line as follows, starting from Straight line Normal form: $$ x\, \cos \alpha + y \sin \alpha = p. \tag{1}$$ ...
2
votes
1answer
26 views

Evaluating area using an integral in polar coordinates

I am trying to find the area of a circle which is given by the polar parameterization $$r(\phi) = \cos\phi + \sin\phi.$$ I can evaluate it in 2 ways and don't know why I get different answers. First ...
1
vote
3answers
347 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\limits_1^2 \int\limits_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 ...
2
votes
0answers
50 views

How to integrate $\frac{1}{\sqrt{x^2+y^2+z^2}}$

want to evaluate $$\int\frac{1}{\sqrt{x^2+y^2+z^2}}dxdydz$$ over entire $\mathbb{R}^3$ except $(0,0,0)$. I did this using polar coordinate and got ...
3
votes
2answers
43 views

Dynamical System transformation

How can the system $$\frac{dx}{dt}=-y+\epsilon x(x^2+y^2)$$$$\frac{dy}{ dt}=x+\epsilon y(x^2+y^2)$$ be transformed into $$\frac{dr}{dt}=\epsilon r^3$$ $$\frac{d\theta}{dt}=1$$ via polar coordinates? ...
1
vote
1answer
35 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
30
votes
4answers
4k 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 = ...
2
votes
0answers
26 views

Repeatedly interpreting polar coordinates as Cartesian

Start with a Cartesian point $(x,y) \in \mathbb{R}^2$, convert it to polar coordinates $(r,\phi)$ ($\phi$ in radians), and then reinterpret $(r,\phi)$ as $(x,y)$, i.e., set $$(x,y) = (r,\phi) \;.$$ ...
1
vote
1answer
1k views

Mass and center of mass of lamina in polar coordinates

I need some help with the following problem which is question number 15.5.4 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the mass and center of mass of the ...
0
votes
1answer
18 views

Finding orthogonal angles / polar coordinate of an n-dimensional vector

Orthogonal angles are the angles used when converting a vector to polar coordinates. So for vector $(1, 1)$, the orthogonal angle is $45$ degrees. Given a vector $(x_1, x_2, x_3, ..., x_n)$, what is ...
3
votes
3answers
39 views

Expressing $ r = \cot(\theta) $ as an equation in terms of Cartesian coordinates $ (x,y) $.

I need to show this equation $r = \cot(\theta)$ as $x$,$y$ using the following laws: $x=r\cos(\theta)$, $y=r\sin(\theta)$ $r^2=x^2+y^2$, $\tan(\theta)=\frac{y}{x}$ This is what I've done : $$r = ...
1
vote
2answers
24 views

How would you convert this particular polar equation to cartesian?

How would you go about converting the polar equation $r^2 = 4cos(2\theta)$ into a cartesian equation in terms of y? I have just started working on polar-cartesian equations, but do not yet have ...
1
vote
2answers
27 views

Help with finding degrees/radians when converting rectangular to polar coordinates?

I was given the problem "determine two pairs of polar coordinates for (-3,0) when theta is greater than 0 degrees and less than 360 degrees" and I know the radii are 3 and -3. When I use arctan 0/-3, ...
5
votes
0answers
50 views

Area of two polar regions

I'm trying to find the region inside r=sinθ and outside r=1+cosθ. My issue is my limits of integration. I get an intersection at $\frac π2$ and one at the pole. What are my limits for the integral? ...
0
votes
2answers
30 views

Caclulate X,Y coordinates of point after rotation around another point of given degrees

There are Two Points A and B. The linear distance between the points is R. I have the ...
1
vote
1answer
16 views

Difference of roots of unity in polar form

I want to write the difference between $n$-th roots of unity in the form $re^{i \theta}.$ It is enough to find the polar form of $1 - \zeta^k$. By thinking geometrically, I can guess the formula $$1 ...
0
votes
1answer
11 views

Find the length of a curve specified by a series of polar co-ordinates.

I have a curve defined by a series of polar co-ordinates, $P_a(r_a,\theta_a)$ through $P_b(r_b,\theta_b)$. I would like to determine the length of this curve. Because the points are from ...
8
votes
3answers
306 views

Laplace's equation in Polar coordinate, an example?

Consider Laplace's equation in polar coordinates $$ \frac {1}{r} \frac {\partial} {\partial r} (r \frac {\partial U} {\partial r}) + \frac {1} {r^2} \frac {\partial^2 U} {\partial \theta^2} = 0$$ ...
2
votes
2answers
8k 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 ...
3
votes
2answers
89 views

show that nth Chebyshev polynomial is an nth order polynomial

Define the Chebyshev polynomial $T_n(x)=\cos(n\cos^{-1}(x)), n\geq 1, T_0=1)$. Show that $T_n(x)$ is an nth order polynomial This is my attempt, however I couldn't reduce it to a polynomial. ...
1
vote
0answers
64 views

Shifting to polar coordinates

Let $\mathbf{r}:[a,b]\to\mathbb{R}^2,\ a,b\in\mathbb{R}, a<b$, be a $C^{\infty}([a,b])$ application (smooth). Is it true that we can always find two functions $r,\varphi :[a,b]\to\mathbb{R}$, $r\in ...
0
votes
1answer
35 views

Point in a spherical triangle test

Given three latitude/longitude coordinates on a sphere forming a triangle, how do I test if a point p is inside that triangle? I know latitude and longitude implies Earth and Earth is not perfectly ...
3
votes
1answer
22 views

Differential length of a logarithmic spiral

I am working on a problem that asks to find the magnetic field at the origin of a logarithmic spiral $r = e^\theta$ from $\theta = 0$ to $\theta = 2\pi$, where the angle $\theta$ is measured ...
2
votes
1answer
37 views

Area under the curve described by θ=ar

I'm interested in finding the area under the curve described by θ=ar, which is a linear curve with slope 'a' in polar coordinates. Here is what the curve looks like: ...
0
votes
0answers
21 views

Evaluating vorticity as a function of velocity components.

So i have the following question.. Consider the axisymmetric flow of a viscous fluid u = ($ \frac{-\alpha r}{2} $, v(r), $\alpha z$) in cylindrical polar coordinates, where $\alpha$ is a positive ...
1
vote
3answers
24 views

Converting unit square domain in (x,y) to polar coordinates

I have the following double integral $\int_{0}^{1}\int_{0}^{1}\frac{x}{\sqrt{x^2+y^2}}dxdy$ The integrand is fairly simple: $\frac{x}{\sqrt{x^2+y^2}}dxdy=\frac{rcos(\theta ...
1
vote
2answers
65 views

Area inside loop of polar equation, unsolvable problem?

Is this problem solvable? "Please find the area inside the first loop of the following equation (using polar coordinates): r = cos$(\theta)$ - sec$(\theta)$." From what I can tell, this function ...