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 ($\theta$).
4
votes
1answer
537 views
Area Bounded by Polar Curves
I am answering sample exams for my Calculus class and my attention was caught by the following item.
Set-up the definite integral or sum of definite integrals equal to the area of the region above ...
3
votes
1answer
388 views
Bézier approximation of archimedes spiral?
As part of an iOS app I’m making, I want to draw a decent approximation of an Archimedes spiral. The drawing library I’m using (CGPath in Quartz 2D, which is C-based) supports arcs as well as cubic ...
1
vote
2answers
628 views
Del operator in 2D polar coordinates
I need to show that the del operator in 2D polar coordinates is $\nabla=e_r\partial_r+\frac{1}{r}e_r+\frac{1}{r}e_{\phi}\partial_{\phi}$. I try the following approach:
...
1
vote
2answers
129 views
Length of a plane curve in polar coordinate
Consider the plane curve $\gamma$ in polar coordinates:
$$
r=r_0+e^{\lambda\theta}, \quad \theta_1 \le \theta \le \theta_2,
$$
where $r_0,\lambda,\theta_1>0$. Is it possible to compute explicitly ...
1
vote
2answers
3k views
area between polar equation $r = \sin\theta$ and $r = \cos\theta$
Below is the exact question and answer from my textbook:
Find the area of the region enclosed between the two curves $C_{1}$
and $C_{2}$ where $C_{1}$ has the polar equation $r = \sin\theta$ and ...
2
votes
3answers
1k views
find all points for intersection between 2 polar equations
I stumped at one of the exercise in my multivariable calculus textbook.
I try to search online but I can't seem to search
on how answer no 3 and 4 below is derived.
I also plot both of polar ...
1
vote
1answer
1k views
Find the equation in polar coordinate form for a straight line through the points with polar coordinates (4,0) and (4,π/3).
Find the equation in polar coordinate form for a straight line through the points with polar coordinates $(4,0)$ and $(4,π/3)$.
Here's my steps:
1.Write the two points in cartesian coordinates: the ...
4
votes
2answers
476 views
Find the area enclosed by the loop $r=2(1-\sin\theta)\sqrt{\cos\theta}$
The diagram shows a sketch of the loop whose polar equation is
$$r=2(1-\sin\theta),\qquad -\frac{\pi}{2}\leq\theta\leq\frac{\pi}{2}$$
a)Show that the area enclosed by the loop is 16/3.
...
0
votes
1answer
310 views
Polar equation of cartesian $y = 1 + 3x$
I have no idea at all what to do on this
I got $$\cos^{-1} \left(\frac{r\sin \theta+1}{3} \right) = \theta$$
Which can be
$$\cos^{-1} \left(\frac{r\sin \left(\cos^{-1} \left(\frac{r\sin ...
0
votes
2answers
122 views
Polar equation of $y = 2$
Maybe I do not understand what is going on here but I cannot get the right answer.
$$y = 2 $$
$$y^2 + x^2 = r^2$$
$$4 + 0^2 = r^2$$
$$ r = 2$$
$$y = r \sin \theta$$
$$1 = \sin \theta$$
$$\theta ...
2
votes
1answer
28 views
gradient in polar coordinate by changing gradient in Cartesian coordinate
I'm tried to do following and I can't see what went wrong.
$$\begin{bmatrix}
\hat r\\
\hat \theta
\end{bmatrix} = \begin{bmatrix}
\cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
...
0
votes
2answers
359 views
Single variable integral to polar coordinates?
I took calculus about 2 semester ago, and I'm trying to brush up on polar coordinates.
I integrated $-x^2+3$ from $x = -\sqrt{3}$ to $\sqrt{3}$ and I got $6.93$
Now I tried to convert it to polar ...
1
vote
1answer
76 views
Maximum point of a polar function
I have a curve C with polar equation
$$r^2 = a^2\cos{2\theta} $$
And I am looking to find the length $x$ when $r=max$
Judging from the equation:
$$r = \sqrt{a^2\cos{2\theta}} $$
R will be maximum ...
1
vote
1answer
197 views
Find the area of the region determined by two curves
Find the area of the region $R$ given by two curves.
So the region $R$ describes the area that is common between the two curves:
$$\begin{align*}
\text{Function 1: } r&= 2\sin(\theta)\\
...
5
votes
2answers
101 views
Why does it always take n numbers to characterize a point in n-dimensional space (or does it)?
I don't know if this is obvious and a dumb question or not, but, here we go. To characterize a point in 2-d space we can use standard $x,y$ coordinates or we can use polar coordinates. There are ...
5
votes
2answers
676 views
Is $r=2\cos(\theta)$ a one-petal polar function?
I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
2
votes
1answer
278 views
How to “rotate” a polar equation?
Take a simple polar equation like r = θ/2 that graphs out to:
But, how would I achieve a rotation of the light-grey plot in this image (roughly 135 degrees)? Is ...
1
vote
1answer
151 views
Fitting of Closed Curve in the Polar Coordinate.
I know how to fit a curve when given some data points in the cartesian coordinate. Recently, I encountered a model that needs to fit a closed curve in the polar coordinate. I'm thinking of deducing a ...
1
vote
3answers
285 views
Polar equation of a circle
A very long time ago in algebra/trig class we did polar equation of a circle where
$r = 2a\cos\theta + 2b\sin\theta$
Now I forgot how to derive this. So I tried using the standard form of a circle.
...
0
votes
2answers
2k views
Set up double integral of ellipse in polar coordinates?
How do you set up a double integral for an ellipse in polar coordinates without using Jacobian or Greens Theorem?
I can't seem to figure out what (or if) the limits of r can possible be.
$x = ...
0
votes
3answers
661 views
Argand Diagram - Quadrants help
I forgot the rules of adding angles when it comes to argand diagrams.
In the first quadrant, you add 90 degrees to whatever angle you get, what about Q2 Q3 Q4 ?
This picture will explain what i mean ...
2
votes
1answer
368 views
Find the Cartesian equation corresponding to $r = \frac{5}{3-2\cos(\theta)}$
Find the Cartesian equation corresponding to $r =
\frac{5}{3-2\cos(\theta)}$
I got it into the form:
$(x^2 + y^2)(3-2x)^2 = 25$
and can see that maybe the equation of a circle will appear, but ...
4
votes
3answers
2k views
Simple proof of integration in polar coordinates?
In every example I saw of integration in polar coordinates the
Jacobian determinant is used, not that i have a problem with the Jacobian,
but I wondered if there's a simpler way to show this which ...
2
votes
1answer
149 views
Find volume of a revolved solid by integrating wedges.
So, lets say that I wanted to find the volume of the solid formed by rotating the area between
$f(x)=\sqrt{1-x^2}, 0<x<1$ and the $x$ axis around the $y$ axis. (This example is simply a ...
2
votes
2answers
115 views
Integral variable substitution using Hausdorff measure
Suppose we have positive density $q$ with "good" qualities (continuity, etc..). I need to calculate this integral:
$$\int_B q(\textbf{z}) d \textbf{z},\ \textbf{z} \in \mathbb{R}^d,$$
where $B \subset ...
1
vote
3answers
176 views
Expressing $e^z$ where $z=a+bi$ in polar form.
I am reading a passage of text that states:
"We can use the fact that $e^{a+bi}=e^a(\cos b+i\sin b)$ has polar form $\left<e^a,b \right>$ to verify that complex exponentials have various ...
1
vote
1answer
136 views
Complex Numbers and polar form
I am given the following information:
$$x[n]= s^n,\qquad n=0,\pm 1,\pm 2,\ldots$$
where $s=\sigma + j\omega = re^{i\theta}$ is a complex number in general.
I was wondering how the following is ...
0
votes
1answer
891 views
How to write parametric equations for a given polar equation?
I'm doing an extra credit problem for math, we haven't learned too much on this topic.
The instructions are: Write parametric equations for the given polar equation.
The problem is: $r = ...
1
vote
1answer
322 views
Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation.
Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation.
Let $t= \theta$, $r=5$, $x=r\cos t$, $y=r\sin t$.
I began with $x=5\cos t-5=5(\cos t-1)$ and $y=5\sin t$. Is that ...
0
votes
1answer
161 views
Finding a geodesic on a plane using polar coordinates
This is from my homework on PDE.
I need to find a geodesic on a plane using polar coordinates. Now, I know
$dl^2 = x^2+y^2$ hence $l=\int \sqrt{dx^2+dy^2}$,
but I get stuck while converting ...
0
votes
2answers
76 views
Laplacian in polar coordinates
I am stuck with an exercise that requires me to find the Laplacian $\Delta u=(D_x^2u+D_y^2u)$ of a 2d-function $u$ in polar coordinates (in the standard Euclidean plane).
I found the following ...
1
vote
1answer
94 views
Explain why for $r=1-a \cos^2(3\theta)$ the leaves have the same size only in the case $a=1$ and $a=2$
Please explain why for $r=1-a\cos^2(3\theta)$ the leaves have the same size only in the case $a=1$ and $a=2$.
Does anyone have an answer to this please?
0
votes
1answer
210 views
Transform integral into polar coordinates
At university we are given a voluntary hand in in the use of maple/matlab, in that
regard I have a double integral I am in dire need to compute, using first cartesian then polarcoordinates.
...
1
vote
1answer
263 views
How do I change a 3D cartesian equation into a polar equation?
I know how to change 2D cartesian equations into polar equations, however I'm having some difficulty with a 3D equation. I am trying to take the cartesian equation x^2+(.75y+4)^2+(z+3)^2=20 and turn ...
0
votes
1answer
49 views
Find complex $z$ such that $z$ has the largest possible real part, and satisfies: $z^7 = -18-18i$
Find complex $z$ such that $z$ has the largest possible real part, and
satisfies the equation:
$z^7 = -18 -18i$
So, the 7th roots of $z = 18\sqrt{2}e^{i\frac{\frac{\pi}{4} + 2\pi k}{7}}$ ...
0
votes
2answers
33 views
Find the Cartestian form of $6 - 7i$ rotated anticlockwise through $\frac{3\pi}{4}$ about the origin
Find the cartestian form of $6 - 7i$ rotated anticlockwise through $\frac{3\pi}{4}$ about the origin
I realize that I am going to be doing something like:
$\sqrt{85}e^{i\alpha}.e^{i\frac{3\pi}{4}}$ ...
1
vote
1answer
909 views
Double Integral, Change of Variables to Polar Coordinates
Quick question on Polar Coordinates.
When evaluating the double integral and changing variables, I'm not sure if the limits are correct.
The question is as follows:
Evaluate $$\int\!\!\!\int_D ...
0
votes
3answers
147 views
Write $\cos(9x)$ in terms of powers of $\cos(x)$ [duplicate]
Possible Duplicate:
How to expand $\cos nx$ with $\cos x$?
Write $\cos(9x)$ in terms of powers of $\cos(x)$
I realize I could solve this by using De Moivre's and binomial expansion:
...
1
vote
1answer
88 views
Given an exact velocity and a “velocity range”, what is the relative velocity range?
I'm trying to calculate the relative velocity ($V_R$) between an exact velocity ($V_0$) and a velocity range ($V_1$).
The exact velocity ($V_0$) is represented simply by ($course$, $speed$).
The ...
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vote
2answers
2k views
Difficult conversion from polar equation to rectangular equation.
How do we convert this into rectangular equation?
$r=5\theta$
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2answers
126 views
How do we get the rectangular form of this?
I know if $\sqrt{x^2+y^2} = x$, then the polar equation of this is $r=cos\theta$
So,how to get the rectangular form of this polar equation, is it complicate:
$r=cos(10\theta)$
5
votes
1answer
605 views
Finding a point on Archimedean Spiral by its path length
I've been curious about Archimedean Spirals and their relations to Sacks Spirals and prime numbers.
I would like to draw some visualizations of the points with a given distance from the center, ...
1
vote
2answers
93 views
Polar Coordinates
It's been ages since i did any coordinate conversions, and typically i have these two which i just can't manage to solve by myself.
I want to express the circle $x^{2}+y^{2}<4, x<0 $
The Area: ...
1
vote
1answer
185 views
Help needed with partial derivatives and polar coordinates, missing term.
I have a missing $\frac{1}{r}\partial_r$ -term (notice the question mark) but cannot see why, could someone hint where I am doing mistake.
2
votes
1answer
238 views
Partial derivatives and orthogonality with polar-coordinates
We are stuck with this question here because I cannot understand the following results. I find it hard to visualize this, let alone deduce from that. How to do it?
Objective to Attack The closely ...
2
votes
1answer
168 views
Explain Dot product with Partial derivatives in Polar-coordinates
Related to page 819 prob 4 in this book. I am incorrectly calculating the left-hand-side (def. LHS), some newbie error with commutativity probably. Ideas?
Errors?
I propose ...
0
votes
1answer
272 views
Orthonormal vectors in Polar coordinates, show $\hat{e}_R=\frac{(x,y,z)}{r}$
Definitions
Unit vector has length 1. Orthonormal vectors are orthogonal and unit vectors.
RobJohn's suggestions for the basis in polar coordinates, here, satisfy the criteria but how can ...
2
votes
0answers
388 views
Explain Triangle perimeter in polar coordinates
The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...
3
votes
2answers
73 views
Using polar form to prove $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$
This was an answer provided to a question I asked previously. I followed the other approaches to the question; however, I couldn't seem to follow this one:
...
2
votes
1answer
44 views
Proving that inversions are isometries with respect to the hyperbolic metric.
I'd like to prove that the standard inversion $$(r,\theta)\mapsto\left(\frac{1}{r},\theta\right)$$ is an isometry with respect to the hyperbolic metric on the upper half-plane, and it would be nice to ...
