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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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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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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137 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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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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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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63 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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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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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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50 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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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 ...
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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 ...
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73 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 ...
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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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$ 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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164 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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61 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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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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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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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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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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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 ...
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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 ...
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68 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 ...
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129 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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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 ...
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78 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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88 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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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 ...
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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 ...
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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 ...
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$\underset{(x,y) \rightarrow (0,0)}{\text{lim}} \frac{xy}{y-x^3}$

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 ...
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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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71 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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18 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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51 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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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? $$\partial_r P= f_r$$ ...
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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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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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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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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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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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How would you find the eccentricity of this conic section?

$4x^2 - 5y^2 - 16x - 50y + 71 = 0$ Thank you!
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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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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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31 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 $$ ...
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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 ...
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Applications of Polar coordinates

What applications exist for Polar coordinates (especially over the more better known Cartesian coordinate system)? Both "applied" applications and applications in pure mathematics may be included for ...
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71 views

Show that a polar equation describes a circle

I want to prove that this polar equation: $$r^2 + 2r(\cos(\theta) - 3\sin(\theta)) = 4$$ describes a circle. I tried converting the equation into a cartesian equation and got $$r^2 + 2x - 6y = 4$$ ...
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63 views

Chain Rule in Polar coordinates

I was looking for an intuitive explanation for the total derivative in polar coordinates. Let me be somewhat more specific: Take a standard line of reasoning that the gradient w.r.t. polar coordinates ...
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42 views

Arc length of polar curve

I was trying to determine the arc length of the polar curve $r = f(\theta) = a(1 - \cos \theta)$, and it was going well until I got to the definite integral. I know that $f'(\theta) = a \sin \theta$, ...