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 ($\theta$).

learn more… | top users | synonyms

1
vote
1answer
56 views

Double integral And polar coordinate system

I have to evaluate this integral over the domain D The Plot would be like this: I decided to use polar coordinate system using it It gives me this but I don't know the upper limit of ...
1
vote
1answer
172 views

Problem plotting hypotrochoids using a computer

I have been trying to use a computer to plot some hypotrochoids, but I've run into some issues. For those that are unfamiliar, the parametric equations of a hypotrochoid are: $$x(\theta) = (R - ...
0
votes
0answers
243 views

Function psi, vector potential, satisfying conditions

Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin ...
0
votes
2answers
1k views

Plotting polar equations of circles not centered at (0, 0)

Good afternoon guys! I'm fairly new to polar coordinates and polar equations, so bear with me please. I understand the equation of a circle with radius $a$ centered at the polar coordinate $(r_0, ...
1
vote
0answers
136 views

Algorithm for finding nearest distance from a point to a curved surface in space

I need to write an algorithm which can find the nearest distance from a point in space to a 3D curved surface which is straight in vertical direction but its projection is an arc of a circle (Similar ...
1
vote
1answer
219 views

Express this polar equation in cartesian form

Having trouble converting this polar equation into Cartesian form: $r = 2 + \sin(\theta)$ This is how far I get: $(r = 2 + \sin(\theta))\cdot r$ $r^2 = 2r + r\sin(\theta)$ $x^2 + y^2 = 2r + y$, ...
2
votes
0answers
239 views

Wrong answer within 'Calculus Solution Manual, Michael Spivak, 3rd ed'

I have a problem with the answer provided in the solution manual of Calculus, Michael Spivak, 3rd ed, The Problem: Consider a hyperbola, where the difference of the distance between the two foci is ...
3
votes
4answers
936 views

Linear combinations of sine and cosine

If you take a linear combination of the cosine and sine function, then the result is again a sinusoid, but with a new amplitude and phase shift. $$a \cos(\theta) + b \sin(\theta) = A \cos(\theta + ...
1
vote
1answer
110 views

Polar equation — find area under graph using double integral

What is the area of the region in the plane bounded by the curve given in polar coordinates $r = 4 + 2\cos(2\theta)$? Could someone walk me through the conversion of polar coordinates to rectangular ...
0
votes
1answer
91 views

What is this called: $ \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = $ … Laplacian?

$ \frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = \left( \frac{\partial^2 f}{\partial r^2} + \frac{1}{r^2}\frac{\partial^2 f}{\partial \theta^2} + ...
2
votes
3answers
64 views

Is the graph of $r^2 = 4$ a circle with radius $2$?

If $r^2 = 4$, taking the square root of both sides will give me $r = 2$, so its graph is a circle with radius $2$. Is this correct? I just wanted to make sure because $r^2$ might imply another graph.
1
vote
3answers
103 views

What is the equivalent polar equation of $x^2 + (y-1)^2 = 1$?

It's a question in the textbook that I have and I am having a hard time understanding it. How am I supposed to get the polar equation with this format?
1
vote
1answer
948 views

What is the cartesian equation of $r = 1 + r \sin(\theta)?$

There are no values given for $r$, or $\theta$. How do I derive the cartesian equation for this? It's a question from a textbook I have.
0
votes
1answer
40 views

A polar integration question

I'm trying to prove this integral $$ \int_0^a \int_0^\sqrt{a^2-x^2} f(x,y) \, \mathrm{d}y \, \mathrm{d}x$$ is the same as $$\int_0^{2\pi} \int_0^a r f(r,\theta) \, \mathrm{d}r \, \mathrm{d}\theta$$ I ...
0
votes
2answers
183 views

Writing a system of ODEs in polar coordinates

I have this system of equations: $$\dot{x}=x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}} \\ \dot{y}=x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}}$$ How can I get this in polar coordinates ? I know that ...
1
vote
1answer
138 views

Using integration and polar coordinates to find the volume of a torus

How would I find the volume of the body formed by revolving the circle $r = f(\theta) = \cos\theta$ about the line $\theta = \frac{\pi}{2}$ ? (This is the circle of radius $1$ centered at $(0,1)$ ...
0
votes
1answer
385 views

What is the graph of $r \cos \theta = 3$?

What is the graph of $r \cos \theta = 3$? I don't get why there is a $\cos \theta$ in the side of $r$, even if I divide both sides by $\cos \theta$, the right side will be $3/\cos \theta$, which ...
0
votes
1answer
51 views

What is the graph of the polar equation $r = e$?

Is it the same as the graph of $y = e$? A straight line?
0
votes
1answer
1k views

How to calculate the polar arc length of the entire cardioid $r=a(1-\cos\theta)$

I'm having a bit of an issue calculating the arc length of $r = a(1-\cos\theta)$. I'll begin by listing the steps I made in my attempt to solve this exercise. We know that the arc length formula ...
1
vote
1answer
41 views

What is the area of the closed curve?

The graph of the polar graph $r=\dfrac{4}{2-\cos\theta}$ forms a closed curve. Find the area of the region inside the curve.
0
votes
1answer
2k views

What is the graph of the polar equation theta = pi?

The question exactly goes like the title. I'm thinking that it's a point on the 3.14, but as I'm typing this I realize that I'm wrong and now I'm out of clues (Google didn't help). Please enlighten ...
0
votes
2answers
101 views

Use Polar Coordinates to Find the Limit…

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geq 0$, $r \to 0^+$ as $(x,y) \to (0,0)$)] $$\lim \limits_{(x,y) \to (0,0)} ...
1
vote
2answers
44 views

Evaluating an integral over inifinty with polars leads to an integral of cosine over inifinity, how can this be resolved?

So I have the integral $$\int_0^\infty\int_0^\infty\frac{yx^2}{x^2 +y^2}e^{-(x^2 +y^2)} \,dx\,dy$$ And converting this into polars gives: $$\int_0^\infty r^2 e^{-r^2}\,dr ...
0
votes
1answer
44 views

What am I missing converting cartesian to polar coordinate system?

I've got the equation $ x^2+y^2=2x $. By looking at the graph of that function, I know that it is equivalent to $ r=2\cos{\theta} $ (graph). However, if I convert it by substituting in using the ...
1
vote
1answer
62 views

Problem with Laplacian while treating polar coordinates as special case of spherical coordinates.

I thought that polar coordinates ($r, \phi$) can be viewed as a special case of cylindrical coordinates ($\rho, \phi, z$) with $z=0$, or as spherical coordinates ($r, \theta, \phi$) with ...
1
vote
2answers
232 views

Converting between polar and Cartesian coordinates

The polar coordinates $r$ and $\varphi$ can be converted to the Cartesian coordinates x and y by using the [[trigonometric function]]s sine and cosine: $$x = r \cos \varphi \,$$ $$y = r \sin \varphi ...
0
votes
1answer
562 views

Finding the area bounded by $r = a(1-\sin\theta)$ and $r = a$

Consider the cardioid $r = a(1-\sin\theta)$ and the circle $r = a$. We have that the cardioid meets the origin at an angle of $\frac{\pi}{2}$, while it reaches its maximum distance from the origin at ...
0
votes
2answers
597 views

Show that the four points given below are the vertices of a rhombus.

Show that the four points, $(5, 8), (7, 5), (3, 5)$ and $(5, 2)$ are the vertices of a rhombus. I tried solving it, by finding out the distances by using the formula $\sqrt{(x_{2}-x_{1})^2 + ...
1
vote
0answers
101 views

Finding intersections points of pairs of polar curves?

Find all intersections of the curves $r=3^{(1/3)}\cos(\theta) , r=\sin(\theta)$ What I have done so far is to just put them equal to each other like this: $3^{(1/3)}\cos(\theta)=\sin(\theta)$, but ...
1
vote
1answer
44 views

Sketch the polar graphs

Any advice on how to sketch polar graphs? I have tried transforming to rectangular coordinates, but its not really much help $$ r=1+\sin(\theta) \\ r^2=4\cos(2\theta) $$ Thanks in advance :)
1
vote
1answer
70 views

transforming cartesian to polar coordinates?

Transform the given polar equation to rectangular coordinates, and identify the curve represented. $$r=\frac{5}{3\sin\theta-4\cos\theta}$$ Any tips? The first thing I tried was replace $\sin\theta$ ...
1
vote
2answers
551 views

Mathematical roses with $4n+2$ petals

In polar coordinates $(r, \theta)$, the equation $$r = \sin\left(a \theta\right)$$ gives a rose with $a$ petals if $a$ is odd, or $2a$ petals if $a$ is even. Thus, the number of petals generated for ...
0
votes
1answer
113 views

Polar to phasor

Let's say that there is a polar equation: -2400 + 8320j To convert this polar equation to phasor form, should the negative be considered when trying to find the angle? Would the angle be +73.91 or ...
2
votes
1answer
53 views

Complex number to polar representation

I'm trying to change the complex number, $-3i$ to polar representation. What I did: $a=0$ $b=-3$ $r=\sqrt{a^2+b^2} = \sqrt{0+3^2} = 3$ $\theta = \frac{b}{a} = \frac{-3}{0}$ But after that I'm ...
0
votes
1answer
85 views

How do i justify integration by polar-coordinates for Riemann-integration?

I completely understand how to transform Lebesgue integration to integration by polar-coordinates using the surface measure. However, i wonder if there is a weaker version of this justifying ...
2
votes
1answer
738 views

Definition of “the surface measure”?

Let $\mu_n$ be the $n$-dimensional Lebesgue measure. Let $||\cdot||$ be a norm on $\mathbb{R}^n$. Define $S^{n-1}=\{x\in\mathbb{R}:||x||=1\}$. I have proven that $\forall A\in\mathscr{B}_{S^{n-1}}, ...
1
vote
1answer
500 views

Find rectangular equation of a cardioid

Given the equation in polar form $$r = 1 - \sin\theta,$$ find the rectangular equation. So far, I found: $$x^2 + y^2 = 1 - 2\sin\theta + \sin^2\theta\quad x = \cos\theta - \sin\theta\cos\theta\quad ...
1
vote
1answer
39 views

How to find polar values of complex number as quick as possible?

I need to calculate these kind of values in exams in best speedy way. Convert $1.46 + 3.17j$ to polar form ($r∠θ$) Is there is any solution to find of the values as quick as possible? By the way, ...
1
vote
1answer
55 views

Reference for studying polar coordinate

There is a theorem about justification of polar-coordinate in Folland-Real analysis p.78. I find it somewhat terse (Maybe it's just me).. I guess this kind of transform is possible even when ...
0
votes
1answer
179 views

Cartesian vector field to vector field

Ok so I have a given vector field in Cartesian coordinates, say \begin{align*} \textbf{v}(x,y)=\frac{dx}{dt}\hat{\textbf{e}}_{1}+\frac{dy}{dt}\hat{\textbf{e}}_{2} \end{align*} Where $dx/dt$ and ...
0
votes
1answer
67 views

Rectangular transformation into Polar coordinates

I was working with a simple transformation of rectangular coordinates - symmetry around the y-axis, i.e. $$f(x,y) = (x, -y)$$ I wanted to express the identical concept in polar coordinates. After ...
1
vote
1answer
58 views

Polar equation and Cartesian equation

For the polar equation, $r \sin \theta = \ln r + \ln (\cos\theta)$ Is that equivalent to $y = \ln x $ ?
1
vote
1answer
57 views

For which $\alpha \in \mathbb{R}$ does $\int_{\mathbb{R}^n} \big(1+|x|\big)^{\!-\alpha} \mathrm{d}x$ exist?

I assume only $\alpha \gt 1$ gives $\int_{\mathbb{R}^n} (1+|x|)^{-\alpha} \mathrm{d}x \lt \infty$ (simply because this is true for $n=1$). I also assume some clever transformation could be used for ...
3
votes
1answer
144 views

Change to polar coordinates when evaluating limits of functions in two variables?

I have a function in two variables $f(x, y)$ and need to calculate the limit $$ \lim_{(x, y) \rightarrow (2, 3)}{f(x, y)} .$$ If I decide to change to polar coordinates, how can I determine where $r$ ...
0
votes
1answer
118 views

Problem understanding solution of complex nth-root of unity

a while ago we had the solution for a complex number task about the nth-root of unity in the complex. But now I am having some difficulties to fully understand it: The task was to find all complex ...
1
vote
3answers
162 views

How to calculate $\theta$ when we know $\tan \theta$.

Hej I'm having difficulties calculating the angle given the tangent. Example: In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this ...
0
votes
1answer
119 views

Velocity of a particle in polar coordinates

The equations $r = 3\sin(2\theta)$ and $\frac{d\theta}{dt} = 2$ describe the motion of a particle in polar coordinates. Find the velocity of the particle in terms of the unit vectors $u_r$ and ...
0
votes
1answer
65 views

Is it possible to change the pole and/or the polar axis in a polar coordinate system?

Citing Wikipedia's article on polar coordinates... In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance ...
1
vote
2answers
778 views

how to find existence and value of limit in multivariable calculus

I was in maths class and i found a question interesting. Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar ...
2
votes
1answer
69 views

Area of a sphere bounded by a paraboloid

I need to find the area of the surface $x^2+y^2+z^2 = a^2$ for $y^2 \ge a(a+x)$. I know that $A = 4a \int_{-a}^0 dx \int_{\sqrt{a^2+ax}}^{\sqrt{a^2-x^2}} \frac{dy}{\sqrt{a^2-x^2-y^2}}$, but I have ...