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
116 views

Double integration in polar coordinates between two circles

I am trying to integrate converting to polar coordinates, between two circles. $$A = \iint_D x \,\mathrm{d}A $$ Ant the domain of integration is set to be the region in the first quadrant between ...
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0answers
40 views

Dot product of gradient and tangent vector

Using polar coordinates with variables $r$ and $\theta$. Let $\vec{r}$ be the position vector. Consider $\nabla \theta \cdot \frac{d\vec{r}}{d\theta}$. This is the dot product of the gradient normal ...
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15 views

Polar Coordinates to Cartesian - Finding Y component

I have the following diagram and frame: I am trying to find out what the equation is that matches XYZ to RThetaPhi. Basically, I need an expression that gives Ys in terms of RThetaPhi. My problem ...
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1answer
81 views

Difference between Euler form and polar / trig form of a complex number

After some readings, I have found out that the difference between the polar / trigonometric form and the Euler form of a complex number consists on the fact that in the first case is expressed the ...
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1answer
26 views

Double integrals in polar coordinates — Multivariable

I've done some research on this topic but I am quite confused about finding the area under a specific volume in polar coordinates. Let's have an example, how would we find the volume of a hyperboloid ...
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2answers
90 views

find the area of the region lying inside the circle $r=6$ and inside the cardioid $r=4-3\sin \theta$.

Well, I drew a graph to visualise it and I found the interceptions $\theta=\arcsin \left(-\frac{2}{3}\right)$. From the graph, by symmetry, I found that the area of region from $\theta$ to $\pi/2$ and ...
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1answer
189 views

Find the area of the region lying outside a circle r=7 and inside the cardioid r=6+7sin theta

So this is the question I have problem dealing with. I know that firstly I need to equate $7$ and $6 + 7\sin \theta$ to get the intersection. And then I am supposed to apply the formula.. But I am ...
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0answers
24 views

Arc Lengths of Points Tangent to a Logarithmic Spiral

Suppose we are given distinct array of $N$ vertices (or Cartesian points) $V_n =(v_1, v_2, ... v_n), v_i \in \mathbb{R}^2$. Taking $v_1$ to be the origin of a logarithmic spiral whose curve ...
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0answers
26 views

Predict a point when you are given initial measurements

From given (x,y) sensor measurements, output by a robot, I need to find robot's heading direction and predict the next location. I have an algorithm that when programmed gives me the correct answer, ...
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0answers
75 views

Tangent of angle between tan line and radial line

How can I use the fact that if the curve whose polar equation is $r=f(\theta)$ is rotated about the pole through an angle $\phi$, then an equation for the rotated curve is $r=f(\theta-\phi)$ to prove ...
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0answers
10 views

Scale factors and metric in cylindrical and spherical coordinates - isotropy of space [duplicate]

In cylindrical (polar) coordinates, the scale factors are $$h_r=1$$ $$h_{\theta}=r$$ $$h_z=1$$ Would it be correct to say that $h_i$ do not depend on $\theta$ because space is isotropic (has the same ...
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0answers
90 views

Scale factors in cylindrical coordinates - geometrical meaning

I am trying to make sense of the scale factors in cylindrical coordinates and their geometrical meaning. To start with something simpler, begin with Cartesian coordinates: $$h_x=h_y=h_z=1$$ One can ...
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1answer
56 views

Integral in n-dimensional spherical coordinates

I have to calculate the following integral: $\int_{B_1(0)} \frac{1}{|x|^m} dx $ where $x \in \mathbb{R}^d$ and $B_1(0)$ is a $d$ dimensional ball around origin with radius equal to $1$. I know I ...
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0answers
45 views

Volume Generated by the revolution of plane figure about the polar axis, with Boundaries formed by Two Polar Curves

The plane figure bounded by the cardioid $r_1=2α(1+cos\ θ)$ and the parabola $r_2=\frac{2α}{1+cos\ θ}$, rotates around the polar axis. Show that the volume generated is $18πa^3$. So the plane i have ...
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2answers
50 views

Show $-27$ in polar form.

A question from my text asks to find the $3$ cube roots of $-27$. The first step in the solution is to immediately show the polar form of $-27$ as $$-27 = 27(\cos \pi + i\sin \pi).$$ Would someone ...
2
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1answer
79 views

Finding the slope of a tangent line to a polar curve at given points

I am given the following polar curve and set of points: $r^2$ = 9cos(2$\theta$) $(0, \frac{\pi}{4})$ $ (0,-\frac{\pi}{4}) $ I need to find the slope of the line tangent to that curve at the given ...
4
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2answers
66 views

Evaluate Integrals by Changing to Polar Coordinates

I'm working on this question for my Calculus III Homework: Evaluate the given integral by changing to polar coordinates. $$\iint_{R} (5x-y)\,dA$$ where R is the region in the first quadrant ...
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2answers
53 views

$ 1 - \cos 2 \Theta$ can be rewritten as $1 - \left( 1 - 2 \sin^2 \Theta\right)$ - I don't understand why though

Going through a video I saw this and wasn't sure how to sort it - given the following : $$ r = 4 \left( 1 - \cos 2 \Theta \right) $$ the part in parenthesis $ 1 - \cos 2 \Theta$ can be rewritten ...
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1answer
192 views

Find the arc length of lemniscate $r=2(\cos(2\theta))^{1/2}$

I have to find the arc length of a lemniscate with polar equation $r=2(\cos(2\theta))^{1/2}$. So far I got like $\sqrt{4\cos(2\theta)+\left(-2\frac{\sin(2\theta)}{\sqrt{\cos(2\theta)}}\right)}$. I ...
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2answers
85 views

How to determine if a point falls on a vector given 2 points and a unit vector. [duplicate]

So I have point A, and point B, with let's say coordinates (1,3,5), and (7,8,9) respectively. Then I have a unit vector C, ...
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1answer
30 views

Can anyone explain these two inconsistent results? Partial derivative calculation.

Let $x=r\cos \theta$ and $y=r\sin \theta$. Find $r_x$. My answer: $r_x=(r_x)^{-1}=x_r=(\cos \theta)^{-1}$. Book answer: $$\frac{\partial (r^2)}{\partial x}=\frac{\partial (x^2+y^2)}{\partial x} \...
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3answers
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Convert $r^2= 9 \cos 2 \theta$ into a Cartesian equation

This is how I tried so far... $r^2= 9 \cos 2 ( \theta)$ $\cos (2 \theta) = \cos ^2 (\theta) - \sin^2 (\theta)$ and $r^2= x^2 + y^2$ so, it will become $x^2 + y^2 = 9 [\cos^2 (\theta) - \sin^2 (\...
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0answers
50 views

How to find intersection of moving circle and line?

Say I have a point, with position (x1,y1) at time t=0, with velocity dx1 and dy1 in the x and y directions respectively, which may or may not collide with a circular entity with radius r, centered at (...
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122 views

Calculating X & Y coordinates of a point offset from an ellipse at a given polar angle and intersecting point

I am using the following equations to identify the x and y coordinates of a point on an ellipse at polar angle θ. $x=\pm\cfrac{ab}{\sqrt{b^2+a^2 (\tan^2\theta)}}$ $y=\pm\cfrac{ab}{\sqrt{a^2+b^2 (\...
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0answers
14 views

Polar plane spiral repitions

I'm just starting out teaching my self about the polar plane using tools like Desmos and have been wondering: When graphing an equation in the polar plane, does it extend forever? All the tools ...
2
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1answer
52 views

cosine of fraction of an angle in terms of the cartesian components

Given, $\cos\theta=\frac{x}{\sqrt{x^2+y^2}}$, how can you write $\cos\frac{\theta}{n}$ (n an integer for simplicity) in terms of x and y? For example, one may say $\cos\frac{\theta}{n}=?\frac{x}{\...
2
votes
1answer
69 views

Why does the polar coordinate method not work?

I tried to calculate the limit $\lim\limits_{(x,y)\rightarrow(0,0)} (x^2+y^2)^x$ By using polar coordinates $ x = r \cdot \cos(\theta)$ $y = r \cdot \sin(\theta)$ resulting in $((r\cdot \cos(\...
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1answer
151 views

how to find the distance between two points in the polar coordinate system?

Help me, please! how to find the distance between two points $ A( x_1,y_1 )$ $ B( x_2,y_2 )$ in the polar coordinate system?
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27 views

How to calculate the integral of a function defined by polar variables?

Let be $f(r,\theta)$ a function defined over a circle of radius $R$ where $0\leq r\leq R$ and $\theta$ is defined as the angle being $0\leq \theta\leq 2\pi$. My question is how to calculate the double ...
0
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1answer
102 views

Translating Polar Functions

How do I rewrite a polar function (expressed in a polar coordinate system $r = F(\theta)$ so the entire curve is shifted right or left $h$ units and up or down $k$ units?
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2answers
90 views

Converting Polar Equation to Cartesian Equation

Heading ##Convert polar equation to Cartesian equation. $$r= \frac{2}{1-\cos\theta}$$ I tried to answer this and this is how I answered it. Please review if it's correct or not. Thank you! :) \...
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0answers
24 views

Perimeter of Overlap of $r_1 = 3+2\cos(\theta)$ and $r_2 = 8\cos(\theta)$

I'm trying to find the perimeter of the overlap of the 2 curves. I started off by finding the points of intersection of the two graphs, getting $(4, \pi/3)$ and $(4, 5\pi/3)$. Here's my integral setup:...
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2answers
49 views

how to compute length in polar coordinates?

The line element $\Delta s^2$ is suppose to be an invariant of Euclidean space. In standard coordinates $\Delta s^2=\Delta x^2+\Delta y^2$ while in polar coordinates $\Delta s^2=\Delta r^2+r^2\Delta \...
2
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1answer
47 views

integral, show identity

let $t>0$. consider the functions $$F(t)=\int_0^{\infty} e^{-tx^2}cos(x^2)\, dx,\quad G(t)=\int_0^{\infty} e^{-tx^2}sin(x^2)\, dx.$$ i want to show that $$F(t)^2-G(t)^2=\frac{\pi}{4}\frac{t}{1+t^...
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1answer
59 views

$\underset{(x,y) \rightarrow (0,0)}{\text{lim}} \frac{xy}{y-x^3}$

Evaluate $$\underset{(x,y) \rightarrow (0,0)}{\text{lim}} \frac{xy}{y-x^3}$$ My attempt: I've tried to use polar coordinates $x=r\cos \theta, \; y = r \sin \theta$: $$\underset{(x,y) \rightarrow (0,...
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0answers
22 views

Segment direction in polar plane

I have the following situation: Base point (green) and segments, for each segment his vertices represented as polar point with theta angle from base point. The problem: For each segment I have his 2 ...
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1answer
78 views

Integral over a solid angle

I've been reading about energy conservation and radiosity from the perspective of computer graphics. The basic idea is simple enough: For all possible incoming light directions $\vec{l}$ and view ...
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1answer
19 views

Curve shape prediction by changing configuration space (Cartesian to polar)

Let us consider the equation $y=3x+2$ which describes a straight line in the 2D Cartesian space. Is it possible to predict the shape of this curve in the polar ($r,\theta$) space? How? I believe that ...
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1answer
56 views

Proving distances of polar coordinates

$r\sin\theta=2, r=\frac{2}{1+\sin\theta}, 0<\theta<\pi$ Line l has the first equation, Curve c has the second. Any point on curve C has polar coordinates (a,$\phi$). The foot of the ...
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1answer
16 views

Line integral of conservative field in polar coordinates

I am solving the vector equation: $$\vec \nabla P(r,\phi) = \vec f(r,\phi)$$ where $\vec f$ is conservative, in polar coordinates. Am I allowed to the following? $$\partial_r P= f_r$$ $$\partial_\...
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5answers
371 views

Finding the orientation of a noisy ellipse

This question comes from a neuroscience study which generates $12$ vectors. The vectors are evenly spaced, $30 n$ degrees for $n=0,\dots, 11$, each with their tail centered on the origin. I am ...
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2answers
63 views

Finding the coordinates of the vertices of an equilateral triangle.

I have an equilateral triangle. I know the orientation of that triangle(that means I know the angle of one of the sides of the triangle with respect to the origin). I know the coordinates of the ...
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1answer
59 views

Gradient of function in spherical coordinates

How do you find the gradient of the function: $$h(r,\theta,\phi) = \frac{1}{r}e^{r^2}$$ I'm not sure what $h(r,\theta,\phi)$ is supposed to output? Is it coordinates? How do you convert this function ...
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2answers
86 views

Why is the graph of $r = a + b\cos \theta$ the same whether a is positive or negative?

So today's lecture was about polar coordinates, and we were taught about the concept up to limacons. I'd like to know why the graph of $r = a + b\cos \theta$ is exactly the same as the graph of $r = -...
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1answer
66 views

double integral over a circular region in polar coordinates

I have a function $f(x,y) = x$, and I want to find the double integral over the circular region $(x-2)^2 + y^2 =1 $ using polar coordinates. Converting the region to polar, we get $r^2 -4cos\theta ...
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1answer
34 views

How would you find the eccentricity of this conic section?

$4x^2 - 5y^2 - 16x - 50y + 71 = 0$ Thank you!
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130 views

Is this a sound demonstration of Euler's identity?

Richard Feynman referred to Euler's Identity, $e^{i\pi} + 1 = 0$ as a "jewel." I'm trying to demonstrate this jewel without recourse to a Taylor series. Given $z = cos\theta + i sin\theta\; |\;|z| = ...
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1answer
64 views

Polar Coordinate System Transform?

What is the fastest way (fewest trigonometric and square root operations) to transform between one radius and angle to that of a polar coordinate system with a different centerpoint? I.e. the polar ...
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1answer
33 views

Solve DOE system with polar coordinates?

I am studying for a exam and one of model questions is solve a DOE system using polar coordinates. I've research and didn't find any reference about this subject. System in question is $$ \left\{\...
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2answers
35 views

Parametric Representation for a Square with Side $1$ Centered at the Origin as a Function of the Angle Measured from the Positive $x$-Axis

While playing with some graphics progamming in OpenGL, I've encounterd this problem: Find the Parametric representation for a square with side $1$ centered at the origin as a function of the angle $...