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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65 views

How to get arc-length of polar function $r= 4(1-\sin{\phi})$?

How can I get arc-length of this polar function? $$ r= 4(1-\sin{\phi})$$ $$-\frac{\pi}{2}\leq\phi\leq\frac{\pi}{2}$$ I know that arc-length of polar function can get calculate by ...
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1answer
742 views

Area between two polar curves $r = 2 \sin\theta$ and $r =2\cos\theta$

I am trying to find the area between the polar curves $r = 2 \sin θ$ and $r = 2 \cos θ$. I set up the area equation as follows: $$\frac12\int_0^{\pi/4}((2\sinθ)^2-(2\cosθ)^2)\,d\theta$$ I could not ...
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2answers
71 views

Complex number polar form equation

I've been struggling with a complex numbers algebra question for a few days now, and the tutor says I still haven't got it right. Express $z_4 =−\sqrt{3} + i$ in polar form. Hence solve the ...
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1answer
46 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 ...
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1answer
107 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 - ...
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234 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 ...
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2answers
371 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, ...
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0answers
85 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 ...
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1answer
149 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: $(r = 2 + \sin(\theta))\cdot r$ $r^2 = 2r + r\sin(\theta)$ $x^2 + y^2 = 2r + y$, ...
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0answers
72 views

Finding an arc-length between 2 points in 3 dimensions

I know how to find an arc-length between two points with coordinates, say $X=(a,b)$ and $Y=(c,d)$. But how do I find the same thing but for, say $X=(a,b,c)$ and $Y=(d,e,f)$? Thanks!
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0answers
180 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 ...
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4answers
569 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 + ...
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1answer
94 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 ...
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1answer
87 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} + ...
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3answers
62 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.
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3answers
73 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?
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1answer
458 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.
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1answer
32 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 ...
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2answers
133 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 ...
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1answer
104 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 $(0,1)$ ...
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1answer
213 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 ...
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1answer
50 views

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

Is it the same as the graph of $y = e$? A straight line?
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738 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 ...
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1answer
39 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.
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1answer
969 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 ...
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2answers
84 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)} ...
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2answers
43 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 ...
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1answer
38 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 ...
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1answer
53 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 ...
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2answers
166 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 ...
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1answer
378 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 ...
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2answers
277 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 + ...
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0answers
55 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 ...
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1answer
38 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 :)
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1answer
58 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 $\sin\theta$ ...
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2answers
206 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 ...
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1answer
54 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 ...
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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 ...
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1answer
67 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 ...
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0answers
27 views

Find the polar equation of a graph / graph a polar equation

What general approach can I take to find the polar equation of a given graph in the polar coordinate system? What general approach can I take to graphing any polar equation in the polar coordinate ...
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1answer
362 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}}, ...
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1answer
199 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 ...
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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, ...
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1answer
42 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 ...
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1answer
123 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 ...
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1answer
54 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 ...
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1answer
55 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 $ ?
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1answer
54 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
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1answer
124 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 $r$ ...
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0answers
407 views

Sketch the polar graph r=2+cos(theta). Find the points of intersection, if any, of this graph with the straight line y=2x-1 (use two decimal places)

I have already sketch the polar graph. and I have to find this graph's intersection point with the straight line y=2x-1 so, I try to solve it like this way: y=2x-1 Rsin(theta)=2Rcos(theta)-1 ...