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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1answer
12 views

Find a vector in cartesian coordinates given its relative location to another vector in spherical coordinates

Here is my problem: -I have an arbitrary normalized vector N in cartesian coordinates -I am trying to find normalized vector M, also in cartesian coordinates -I am given the azimuth and polar ...
3
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1answer
23 views

Help solving an ODE

This is an example in my book. It is for the following system: \begin{align*} x'&=y+x(1-x^2-y^2)\\ y'&=-x+y(1-x^2-y^2) \end{align*} So using polar coordinates we get the following system ...
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1answer
19 views

How to calculate Polar coordinates for Complex Polynomials of Higher Degree?

When such I have a complex number such as $3 - 4i$, I can calculate the $r$ with $r=\sqrt{X^2+Y^2} = \sqrt{3^2+4^2}$. But how do I solve this when I have a complex number such as $(2+6i)^6$
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1answer
21 views

Using Polar Integrals to find Volume of surface

Here's the Question and the work that I've done so far to solve it: Use polar coordinates to find the volume of the given solid. Enclosed by the hyperboloid $ −x^2 − y^2 + z^2 = 61$ and the plane $z ...
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0answers
14 views

Using polar form to show that a simple critical point is a spiral point

This is the question in my "homework." I say "homework" because it is not picked up or graded but we are supposed to do it for practice, anyhow here's the question: Given the system ...
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2answers
39 views

What is the Cartesian form of $r = \dfrac2{1 + \sin \theta}$? [on hold]

What is the Cartesian form of the polar equation $r = \dfrac2{1 + \sin \theta}$?
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1answer
23 views

Changing the domain of integral

I am studying how we use polar substitution to solve double integrals. However, I am struggling with finding the correct limits of the transformed integrals to obtain a suitable solution. eg: Why ...
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0answers
15 views

Question concerning the domain of polar coordinate.

So in the problems I encountered, I find it confusing about the domain of $\theta$. Problems take the form: For arbitrary function $f(x,y)$, and $$\displaystyle \iint_S f(x,y)dxdy=\iint_T ...
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1answer
18 views

Sinusoids closed under addition, Euler's Formula

Real sinusoids with the same frequency are closed under addition. If $$f(\omega) = A_1 \cos(\omega + \phi_1) + A_2 \cos(\omega + \phi_2)$$ Then there is some $A_3$ and $\phi_3$ so that: $$f(\omega) ...
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1answer
15 views

For a 2 variable function, are there conditions that guarantee you can verify a limit by using only straight line trajectories?

So a recent post gave a nasty 2-variable function: $$f(x,y) = x^2y/(x^4+y^2)$$ and after changing to polar coordinates, you get that the limit is always equal to zero if you hold $\theta$ fixed and ...
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2answers
47 views

$r=5 \sec(\theta)$ into rectangular

I need to convert $r=5\sec(\theta)$ into rectangular form. I think I need to multiply both sides by $r$, because $r^2 = x^2 + y^2$, but i'm not sure how to convert $r=5\sec(\theta)$ in terms of $x$ ...
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1answer
38 views

What is the length of the cardioid $r=1-\cos(\theta)$?

I know generally how to solve this problem and was able to solve it a week or so ago. However, I keep getting stuck when trying to find $dr/d\theta$. I know that it should simplify to $-\sin(\theta)$ ...
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0answers
31 views

Shifting a plot in polar coordinates

Say we have the plot of a function $r=f(\theta)$ and want to "relocate" it to $(h,k)$. Is there a general procedure for this? I have tried the following tactic to no avail on the following example: ...
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1answer
25 views

Trigonometry for steradian angle

Polar coordinate system is very closely associated with trigonometry. For instance, given an angle in radian, we can find its corresponding 2-D cartesian coordinates using ...
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1answer
30 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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0answers
34 views

Area between two polar curves r = 2 sin θ and r = 2 cos θ

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
36 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
21 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
40 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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1answer
177 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
30 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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1answer
21 views

Express in polar form, $Z=0-j5$. [closed]

Express in polar form, $Z=0-j5$ $j=i$
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0answers
24 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
45 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
14 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
39 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
56 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
29 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
62 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
60 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
45 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
52 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
26 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
52 views

System of equations in polar coordinates $\dot{x}=x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}} \\ \dot{y}=x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}}$

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
32 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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0answers
58 views

How to teach polar integrals

Based on Calculus II calendars everywhere, apparently polar area integrals are something we expect freshmen to fully grasp after one single lecture. (Or even less: the one single lecture is often ...
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1answer
53 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
43 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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1answer
78 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
30 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
55 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
24 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
38 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
25 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
20 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
62 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
103 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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0answers
25 views

Finding Polar Change Of Coordinates

I have a camera with pan/tilt motors. I want to point it at an object with a known pan (θ) and tilt(ϕ) from the camera's location. If the camera is mounted on a level base, this is trivial, I just set ...
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2answers
40 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
15 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 ...