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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Finding the polar form of a complex number

I have the following complex numbers : -3,18 +4,19i I can calculate $r=\sqrt{a^2+b^2}$ Which gives r=5,26 now I know that cos $\theta = \frac{a}{r}$ gives $\theta=127,20$ degrees But when I do ...
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2answers
58 views

Double integral with Polar coordinates - hard example

Calculate using polar coordinates: $$\iint_{D}^{} (x^2+y^2)^\frac{1}5 \ dx \ dy $$ where D is the region inside the circle with radius 1. Working: D: $ \ x^2+y^2=1 \\ $ so $ 0 \leq r \leq 1 \ \ , $ ...
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Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
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1answer
27 views

Finding the area of a polar region

I am trying to find the area inside the curve $$ r = 2 + \sin2\Theta + \cos3\Theta .$$ It's a very weird looking function after graphing, and I'm not quite sure how I'm supposed to proceed. There's ...
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26 views

How to change the equation to polar form?

Compute $\displaystyle\int^\infty_{-\infty} dx\displaystyle\int^\infty_{-\infty} dy\displaystyle\int^\infty_{-\infty} dz \delta\left(\sqrt{x^2 +y^2+z^2} - R\right)$.
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29 views

Parametrization of a rotating surface

What is the parametrization of a surface obtained by rotating the circle $(y − 3)^2 + z^2 = 1, x = 0$ about the z-axis. I came up with the parametrization $S(r,θ) = (r , 3+cosθ , sinθ)$, is it ...
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2answers
16 views

How to calculate polar angle of point given a reference point?

I want to calculate polar angle of some points based on different reference points. Usually polar angle is calculated based on reference point (0,0). What is the procedure to calculate polar angle ...
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1answer
49 views

Simple proof of the Cauchy-Crofton formula on the sphere?

Let $\gamma$ be a regular curve on the sphere. In a lecture, the following result was used $$L(\gamma)=\frac 14 \int_{S^2} \sharp (\gamma \cap \xi ^\perp)d\xi$$ $\xi^\perp$ is the plane with normal ...
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36 views

Does a plane curve with polar equation $r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$ have a name?

Does a plane curve with polar equation $$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$$ where both $\lambda_i>0$ have a name? It's very similar to hippopede, also known as lemniscate of Booth, ...
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24 views

Integral that results in the fraction of two gamma functions

I'm trying to show this equation $$ \int\limits_0^\infty \mathrm{d}x_1 \dots \mathrm{d}x_n \left( 1 - \sum_{i=1}^n x_i \right)^k \Theta \left( 1 - \sum_{i=1}^n x_i \right) = ...
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19 views

Parametric Equation of conics: Parabola

Let $P(ap^2,2ap)$ and $Q(aq^2,2aq)$ be two points on the parabola $y^2=4ax$ such that PQ is the focal chord. Let $A(at^2,2at)$ and $B(as^2,2as)$ be two other variable points on $y^2=4ax$. a) Show ...
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1answer
39 views

Double checking a change of variables to Polar in a differential equation

I have $$\frac {\mathrm{d}x}{\mathrm{d}t}=x-y-(2x^2+y^2)x $$$$\frac{\mathrm{d}y}{\mathrm{d}t}=x+y-(x^2+2y^2)y$$ I have calculated $\frac{\mathrm{d}r}{\mathrm{d}\theta} ...
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33 views

Angle using cross product

I have a situation. Please refer to a figure below: I have r1, r2, Ɵ1, Ɵ2 as well the reference line. I want to find out angle Φ(phi). i.e. (angle PBA). Edit_1 The link provided solves the ...
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2answers
88 views

Find the volume of the solid bounded above by the cone $z^2 = x^2 + y^2$, below by the $xy$ plane, and on the sides by the cylinder$ x^2 + y^2 = 6x$.

Q: Find the volume of the solid bounded above by the cone $z^2 = x^2 + y^2$, below by the xy plane, and on the sides by the cylinder $x^2 + y^2 = 6x$. I can't figure out what I'm doing wrong in my ...
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50 views

How is double integral variable substitution different from one variable trigonometric substitution?

I'm studying variable change in double integrals and I understood the reasoning behind the formulas as described really well here. However, geometric arguments for analysis don't convince, as well as ...
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1answer
27 views

Convert Cartesian function to polar function

A problem on my math homework is $x = -4$ convert to a polar function. What are the steps, the examples in my book are only for $y=x$ functions.
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1answer
44 views

Plotting a exponential form of complex number over an angle on an Argand Diagram

Say I had to plot the expression $$\frac{\pi e^{i\theta }}{4\theta}$$ where $\frac{\pi}{4} \le \theta \le \frac{9\pi}{4} $ on an Argand diagram, how would one go about doing so? If it was just the ...
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12 views

Domain in polar coordinates with a square and a discus

I was doing some studying in Steward's Calculus when I came onto this problem. I am asked to integrate a certain function $f(x,y)$ in this domain. I know how to do it when the inner boundary is a ...
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36 views

Polar coordinates for double integral for $\theta$

To evaluate the integral $$\iint_D \sqrt{x^2+y^2}dA$$ $$D=\{(x,y)\mid0\leq(x-1)^2+y^2\leq1 \}$$ it should be best to change variables into polar coordinates to get ...
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1answer
38 views

Calculate surface area of flat figure by using double integral and polar coordinates

Check me please. I tried check it via WolframAlpha, but I don't trust in it 100%. Task: Calculate surface area of flat figure by using double integral in polar coordinates. Figure confined by line: ...
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13 views

Derivation Check: Point Described by Cylindrical Coordinates to Euler Angles

this post is quite long so thank you in advance for those who get through it. I made this post as a confirmation of my logic so others can check over my work and since I didn't find any posts relating ...
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35 views

Calculate double integral using Polar coordinate system

Need to calculate $\int_{0}^{R}dx\int_{-\sqrt{{R}^{2}-{x}^{2}}}^{\sqrt{{R}^{2}-{x}^{2}}}cos({x}^{2}+{y}^{2})dy$ My steps: Domain of integration is the circle with center (0,0) and radius R; $x = ...
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Evaluating $\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz$ between balls in $\Bbb R^3$

I am working on the following problem: Evaluate: $$\iiint_R \log\Big((x^2 + y^2 + z^2)^\frac{3}{2}\Big)\, dx\ dy\ dz,$$ where $R = \big\{(x, y, z) : 1 \leq x^2 + y^2 + z^2 \leq 2^2 \big\}$ is the ...
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1answer
31 views

Relating $dS$ and $d\theta$ for computation of line integrals

I'm asked to compute $$\int_C \vec{F} \cdot d\vec{s}$$ where $\vec{F} = A_0(x\hat{y} - y\hat{x})$, along a circular path, counterclockwise, about the origin with radius 4. I begin by writing $$ ...
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1answer
38 views

What are the characteristics of functions that look the same in both polar and rectangular graph?

Today, I am doing practice for SAT. In a textbook example, I see $$r=\frac{1}{\sin\theta}$$ My textbook is telling me that this particular function looks the same whether it's graphed on a polar or ...
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1answer
64 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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33 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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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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48 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
58 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
68 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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18 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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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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33 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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46 views

Find the branch function for $z^{\frac{1}{2}}$

Find the branch function for $z^{\frac{1}{2}}$ and using Cauchy-Riemann equation to prove that the branch functions are analytic for all z in Complex except along the branch cut. I did $z = ...
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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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54 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
37 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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30 views

Converting double integral to polar coordinates

I dont get how $dxdy$ become $rdrd$$\theta$ in this computation Instead, shouldn't it be $rdr$ $\cos^2\theta$?
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47 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
54 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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51 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
113 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
42 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
23 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
279 views

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
38 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 ...