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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Converting coordinates

I am having huge trouble converting between cartesian to polar and to spherical. I need these methods for my limits of integration in most cases but using the formulas never seem to work, I just ...
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2answers
22 views

parametric polar equation of a circle

I discovered that Mac's Grapher has a parametric polar mode, i.e. where $r$ and $\theta$ can be specified in terms of a parameter, usually $t$. I am attempting to convert the generic equation for a ...
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2answers
117 views

How can $e^a = 0$ when integrating?

I was doing a question which asked me to turn the following into polar equation:$$(y + x − x(x^2 + y^2))\frac{dy}{dx} = y − x − y(x^2 + y^2)$$ With a lot of messy algebra I can get it down to: ...
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5answers
46 views

Finding a Coordinate on a circle using radius, angle, and origin

I am trying to calculate a point on a circle using an angle and a different point. With this picture, I know the origin O, the radius r, the angle A, and the point B. Now I want to find the point ...
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12 views

Geometric or arithmetic sequences in polar coordinates

While studying the properties of the logarithmic spiral, I came across a theorem stating that secant lines drawn from the origin (pole) to the spiral form a geometric progression. I was interested to ...
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1answer
24 views

Polar equations (further question)

Consider the polar equations of the form $r=a\cos(b\theta)$ and $r=a\sin(b\theta)$. What is the nature of the graphs of these two polar equations and then summarize some generalizations with respect ...
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1answer
46 views

Why does $\sin\phi=r\frac{d\theta}{ds}$ and $\cos\phi=\frac{dr}{ds}?$

The relation between $p$ and $r$ where $p$ is the length of the perpendicular from the fixed point $O$ on the tangent to the curve at any point $P$ is called pedal equation of the curve. I want to ...
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1answer
32 views

Integrating using polar co-ordinates

Hey I've just finished an exam paper and just am stuck with one question. It's something that usually makes sense to me but for some reason I can't get this one: Let $R = \{(x,y) : x,y ≥ 0, x^2 + ...
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22 views

Converting a vector $v \in \mathbb{R}^2$ given in Polar coordinates to Cartesian coordinates

I know that switching inbetween Polar coordinates and Cartesian coordinates in $\mathbb{R}^2$ can, on suitable open subsets of $\mathbb{R}^2$, be done via $(x, y) = (r cos \theta, r sin \theta)$. Let ...
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3answers
34 views

Integral conversion to polar coordinates - bounds

I have an integral $$\int_0^1 \int_0^1\sqrt{x^2+y^2}\ dxdy $$ and its result is $\approx0.765...$ I convert it to polar coordinates and get $$\int_a^b \int_c^dr\ drd\phi $$ But how can i compute ...
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1answer
20 views

Calculating the area between a petal and a circle (Polar Coords)

The question states: The graph of $2\cos(3\theta)$ has three petals (also called "leaves" or "lobes"). The intersection of one of those petals with the circle $r = 1$ is shaded in the figure. ...
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2answers
61 views

Sketch the heart and indicate its orientation with arrows $ r = 1 - \cos(\theta)$. Find the area enclosed by the heart

Hi all I am trying to figure out how to sketch the heart. Here is what I have tried so far: $$r = 1 - \cos(\theta) \\ r(r = 1 - \cos(\theta)) \\ r^2 = r - r\cos(\theta) \\ $$ Use the fact that $$r ...
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1answer
49 views

The Laplacian in Polar Coordinates

When solving the Laplacian in polar coordinates, $x=r\cos\theta$ and $r^2=x^2+y^2$. When finding $\frac{\partial u}{\partial x}=\frac{\partial u}{\partial r}\frac{\partial r}{\partial ...
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1answer
25 views

Raising Polar to the Power

Ok guys, I'm back! This is gonna be my last question here, as I have practically finished test review. Raise to the the power: b ) $(8 cis(120 ^\circ)))^{1/3}$ ... Well for this there are two ...
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4answers
100 views

Find the two square roots of $i$

I have this question I am stumped upon for my Test-Review: Write $i$ as a complex number in polar form. Use the result and DeMoivre's Theorem to find the square roots of $i$. I got the first ...
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3answers
54 views

Converting into rectangular form

I have 2 related questions: First: Let $z_1 = 2+2i$ and $z_2 = 2-2i$. Find $z_1z_2 $ in rectangular form. I have no idea... I'm also clueless about this question: Change the following to ...
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1answer
37 views

Find the area bounded by $r=6\sin(2\theta)$

Winplot plot: I tried this: $$A = 4 \cdot \frac 1 2 \int_0^{\pi/2} (6\sin(2\theta))^2 d\theta$$ Is that right? How about $$A = 8 \cdot \frac 1 2 \int_0^{\pi/4} (6\sin(2\theta))^2 d\theta$$
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1answer
24 views

Find the area inside $r=2+2\sin(\theta)$ but outside $r=4\sin(\theta)$.

Winplot plot: I tried this: $$A = 2 \frac 1 2 \int_0^{\pi/2} (2+2\sin(\theta))^2 - (4\sin(\theta))^2 d\theta + 2 \frac 1 2 \int_{\pi}^{3\pi/2} (2+2\sin(\theta))^2 d\theta$$ Is that right? How ...
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0answers
34 views

Find the volume(solid) , transform rectangular function to polar function

if want find volume of this problem Under $ f(x)=x^2+y^2-4 $ and inside $ x^2 + y^2=9$ in plane $z=0$ Can I use this integration in polar functions? $$\int_0^{2\pi} \int_2^{3} (r^2 - 4) r ...
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1answer
45 views

Converting to polar coordinates in integral over $\mathbb{R}^{n}$

I have an integral equation $$\int_{\mathbb{R}^{n}}F(x)dx=1$$ for $F(x):=\frac{C}{(1+|x|^{2})^{\frac{n+1}{2}}}$. And I want to convert the above integral equality to polar coordinates. Am I right in ...
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1answer
25 views

Converting Polar Coordinates to Regular Coordinates

If you guys didn't know, I have my quiz tomorrow and I have one last thing to ask to this Community! I am completely confused on how to convert polar coordinates to regular coordinates. The teacher ...
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5answers
25 views

Labeling Negative Polar Coordinates

i have a quiz coming up tomorow and i have this major question.. its a question about how to plot it exactly when it is negative. Let me go through the whole set: 1) ($4$ , $60^\circ$) 2)($-4$ , ...
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1answer
29 views

Triple Integration of gravitational potential

Integrate $\int_{0}^{2\pi } \int_{0}^{a} \int_{0}^{\pi/2} \frac{ G\rho r^2 sin \theta}{ {(r^2-2rt cos \theta + t^2 )}^{\frac{1}{2}}} d \theta dr d \phi$, where $\rho, t $ are constants. Sorry ...
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1answer
34 views

Changing a rectangular equation into polar equation

Find the polar equation that has the same graph as the given rectangular equation. I have used the method of substituting $r \sin θ$ for $y$ and $r\cos θ$ for $x$. $x^2=8(2−y)$ ...
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2answers
59 views
2
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1answer
38 views

Rectangular coordinates transform in polar coordinates $\displaystyle\int_{0}^2\int_0^{x} f(x,y) \,d y\,d x$

If i have this integral in rectangular coordinates and i want to transform in polar coordinates $$\int_{0}^2\int_0^{x} f(x,y) \,dy\, d x$$ The limits in $\displaystyle 0\le\theta\le\frac\pi4 $ but ...
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1answer
63 views

Find the area inside the lemniscate $r^2 = 8 cos 2\theta$ and outside the circle $r = 2$.

Fooplot graph: I think the formula is $$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$ where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$. This ...
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0answers
27 views

Find the area inside the circle $r = 10 \sin \theta$ and above the line $r = 2 \csc \theta$.

Fooplot graph: I think the formula is $$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$ where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$. This ...
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Find the area common to the inside of the cardioid $r = 1+\sin \theta$ and the outside of the cardioid $r = 1 + \cos \theta$.

Fooplot graph: I think the formula is $$A = \frac 1 2 \int_{\alpha}^{\beta} (\text{outer})^2 - (\text{inner})^2 d\theta$$ where $\alpha, \beta$ are where they intersect in $[0, 2\pi]$. This ...
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1answer
39 views

Find the area of the small loop of the limacon $r = 1+2\cos(\theta)$

Find the area of the small loop of the limacon (graph): $$r = 1+2\cos(\theta)$$ What I tried: Set $r=0$ to get $\theta = 2\pi/3, 4\pi/3$. Then $$A = \frac 1 2 \int_{2\pi/3}^{4\pi/3} r^2 ...
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1answer
36 views

$r = \sin(3\theta)$ Manual graph

What is the thought process behind the second line of the following, namely that the sin of 3 theta equals +/- 1? Thank you. $r = \sin(3\theta)$ $\sin3\theta = +- 1$ $3\theta = ...
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2answers
26 views

Find the polar equation of the circle with center on the line theta = pi, of radius 1, and passing through the origin. [closed]

Question Find the polar equation of the circle with center on the line theta = pi, of radius 1, and passing through the origin. i set a point (a,pi) on the line and going to this equation a = ...
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Infinitesimal area element in polar coordinate

We know, that the infinitesimal area element in Cartesian coordinate system is $dy~dx$ and in Polar coordinate system, it is $r~dr~d\theta$. This inifinitesimal area element is calculated by measuring ...
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1answer
27 views

Convert ODE to polar coordinates.

$$k \frac{d}{dx}[A(x)\frac{dT(x)}{dx}] - hP(x)[T(x) - T] = 0 $$ What I had in mind was: $$x = rcosϴ, r = \frac{x}{cosϴ} , \frac{dr}{dx} = \frac{1}{cosϴ} $$ $$\frac{dA(x)}{dx} = ...
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2answers
20 views

Polar Coordinate Conversion (Integration)

I want to convert some integrals to use polar coordinates as my differentials, my problem is getting the limits. So this is the first concept I am not understanding: If I have a circle in the ...
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default Metric Unit Circle in plane with polar coordinates - NAME

Quick question: Is there a certain name for this metric? $d(\theta_1,\theta_2) = \left\{ \begin{array}{ll} |\theta_1-\theta_2| & \mbox{if } |\theta_1-\theta_2|\le \pi \\ ...
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0answers
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Direction of a ray in the hemisphere

If my surface has normal (0,0,1), and I center a hemisphere about that normal, how do I compute the ray that is cast in direction $[\theta, \phi]$ within that hemisphere?
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1answer
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Rectangular to polar conversion

I am trying to write this fraction in polar form (4+10i)/(24i-5) . I am having trouble to get the angle of the polar conversion. I know that in order to get the angle I need to write ...
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3answers
42 views

Polar coordinate form of circle equation

I am trying to convert the equation $$(x-0.9)^2+y^2 = 0.1^2$$ into polar coordinates. Using $x = r\cos \theta$ and $y = r\sin \theta$ I get that $$ r = \frac{1.8 \cos\theta \pm ...
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1answer
41 views

Convert Integral Rectangular to Polar

How can convert this problem $$ \int_0^2 \int_x^\sqrt{8-x^2} \left(x^2+y^2\right)^{3/2} dydx $$ I convert limits and funtion to polar cordinates as follows: $$ \begin{split} r^2 &= ...
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2answers
24 views

How to express two variables in two other variables

If: $A=R\cos x$ and $B=R\sin x$ Then how can I express $R$ and $x$ in terms of $A$ and $B$ in a rigorous way? Meaning that I take the domain and range in account? I tried: $$\cos x=\frac{A}{R}$$ ...
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1answer
28 views

How to change complex numbers into polar form? [closed]

How do I changecomplex numbers, for example $2+3i$ to polar form of $re^{i\theta}$. Thank you for any answers.
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2answers
25 views

Converting specific equations from Polar to Cartesian

These different equations are given in Polar and my goal is to plot them in Cartesian coordinate system: $r = \cos(4φ)$ $ φ = \dfrac r {r-1}$, $r > 1$ I am aware of: $x = r \cos( φ )$ $y = r ...
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0answers
33 views

Transform system to polar and sketch phase portrait. Show that $(0,0)$ is an unstable focus.

Transform the system $$x' = y - x(x^2+y^2-1)$$ $$y' = -x - y(x^2+y^2-1)$$ to polar coordinates, and sketch the phase portrait. Show that it has a unique limit cycle and that all ...
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2answers
18 views

How can I find the limits of this iterated polar integration?

How can compute the area of the triangle whose corners are at the origin, (1,0) and (1,1). I solved this with r integral first but I could not find the correct limits for theta integral first order. ...
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2answers
43 views

Polar equation of the curve y = sinx

I am looking for the polar equation of the following curve given in Cartesian Coordinates. y = sinx Any kind of hint or help is appreciated.
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2answers
25 views

Quick Question regarding De Moivre's Theorem

If $z^2 = 2 - 2i$ find z using the theorem of De Moivre For this question, i first expressed it in polar form which is $$2\sqrt{2}\left(\cos{\frac{7\pi}4} + i\sin{\frac{7\pi}4}\right)$$ Now because ...
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0answers
24 views

Finite difference of radial Laplace operator doesn't give a symmetric (hermition in general) matrix

I'm using the central difference to convert the radial part of Laplace operator into a matrix. $\nabla^2 u = \frac{\partial^2 u}{\partial r^2}+$ $\frac{1}{r}$ $\frac{\partial u}{\partial r}$ which ...
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0answers
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Proof of alternate cartesian to polar transformation of theta

My vector calculus lecturer has claimed that rather than the angle $\theta$ in the transformation from cartesian coordinates $(x,y)$ to polar coordinates $(r,\theta)$ can not only be given by: $$ ...
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1answer
33 views

Uniform Convergence: Poisson Kernel

If we fix $θ_∗ > 0$, then $P(r, θ) → 0$ uniformly on the set $ \left\lbrace θ : |θ| ≥ θ_∗ \right\rbrace $ as $r → a^-$ $$P(r,\theta) = \frac{a^2-r^2}{a^2-2r\cos(\theta)+r^2}$$ $0\leq r ...