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
847 views

How do I change a 3D cartesian equation into a polar equation?

I know how to change 2D cartesian equations into polar equations, however I'm having some difficulty with a 3D equation. I am trying to take the cartesian equation x^2+(.75y+4)^2+(z+3)^2=20 and turn ...
0
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1answer
60 views

Find complex $z$ such that $z$ has the largest possible real part, and satisfies: $z^7 = -18-18i$

Find complex $z$ such that $z$ has the largest possible real part, and satisfies the equation: $z^7 = -18 -18i$ So, the 7th roots of $z = 18\sqrt{2}e^{i\frac{\frac{\pi}{4} + 2\pi k}{7}}$ ...
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2answers
62 views

Find the Cartestian form of $6 - 7i$ rotated anticlockwise through $\frac{3\pi}{4}$ about the origin

Find the cartestian form of $6 - 7i$ rotated anticlockwise through $\frac{3\pi}{4}$ about the origin I realize that I am going to be doing something like: $\sqrt{85}e^{i\alpha}.e^{i\frac{3\pi}{4}}$ ...
2
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1answer
2k views

Double Integral, Change of Variables to Polar Coordinates

Quick question on Polar Coordinates. When evaluating the double integral and changing variables, I'm not sure if the limits are correct. The question is as follows: Evaluate $$\int\!\!\!\int_D ...
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3answers
232 views

Write $\cos(9x)$ in terms of powers of $\cos(x)$ [duplicate]

Possible Duplicate: How to expand $\cos nx$ with $\cos x$? Write $\cos(9x)$ in terms of powers of $\cos(x)$ I realize I could solve this by using De Moivre's and binomial expansion: ...
1
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1answer
126 views

Given an exact velocity and a “velocity range”, what is the relative velocity range?

I'm trying to calculate the relative velocity ($V_R$) between an exact velocity ($V_0$) and a velocity range ($V_1$). The exact velocity ($V_0$) is represented simply by ($course$, $speed$). The ...
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2answers
4k views

Difficult conversion from polar equation to rectangular equation.

How do we convert this into rectangular equation? $r=5\theta$
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2answers
200 views

How do we get the rectangular form of this?

I know if $\sqrt{x^2+y^2} = x$, then the polar equation of this is $r=cos\theta$ So,how to get the rectangular form of this polar equation, is it complicate: $r=cos(10\theta)$
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1answer
1k views

Finding a point on Archimedean Spiral by its path length

I've been curious about Archimedean Spirals and their relations to Sacks Spirals and prime numbers. I would like to draw some visualizations of the points with a given distance from the center, ...
2
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2answers
127 views

Polar Coordinates

It's been ages since i did any coordinate conversions, and typically i have these two which i just can't manage to solve by myself. I want to express the circle $x^{2}+y^{2}<4, x<0 $ The Area: ...
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1answer
276 views

Help needed with partial derivatives and polar coordinates, missing term.

I have a missing $\frac{1}{r}\partial_r$ -term (notice the question mark) but cannot see why, could someone hint where I am doing mistake.
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1answer
387 views

Partial derivatives and orthogonality with polar-coordinates

We are stuck with this question here because I cannot understand the following results. I find it hard to visualize this, let alone deduce from that. How to do it? Objective to Attack The closely ...
2
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1answer
184 views

Explain Dot product with Partial derivatives in Polar-coordinates

Related to page 819 prob 4 in this book. I am incorrectly calculating the left-hand-side (def. LHS), some newbie error with commutativity probably. Ideas? Errors? I propose ...
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1answer
456 views

Orthonormal vectors in Polar coordinates, show $\hat{e}_R=\frac{(x,y,z)}{r}$

Definitions Unit vector has length 1. Orthonormal vectors are orthogonal and unit vectors. RobJohn's suggestions for the basis in polar coordinates, here, satisfy the criteria but how can ...
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0answers
959 views

Explain Triangle perimeter in polar coordinates

The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...
3
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2answers
87 views

Using polar form to prove $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$

This was an answer provided to a question I asked previously. I followed the other approaches to the question; however, I couldn't seem to follow this one: ...
2
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1answer
66 views

Proving that inversions are isometries with respect to the hyperbolic metric.

I'd like to prove that the standard inversion $$(r,\theta)\mapsto\left(\frac{1}{r},\theta\right)$$ is an isometry with respect to the hyperbolic metric on the upper half-plane, and it would be nice to ...
2
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1answer
3k views

Transforming the Laplace operator from Polar to Cartesian coordinates

I'm trying to find the error in my logic here. Let's say we are given the Laplace operator in polar coordinates: $$ \frac{\partial^2}{\partial r^2} + \frac{1}{r}\frac{\partial}{\partial r} + ...
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0answers
876 views

Conversion of motion equation from Cartesian to Polar coordinates: Is covariant differentiation necessary?

Say I have the following equation of motion in the Cartesian coordinate system for a typical mass spring damper system: $$M \; \ddot{x} + C \; \dot{x} + K \; x = ...
2
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2answers
204 views

How do I find the limit of $\frac{xy\sqrt{|xy|}}{x^2 + xy + y^2}$ as x and y approach zero?

I am trying to find: $$\lim_{(x,y)\to (0,0)}\frac{xy\sqrt{|xy|}}{x^2 + xy + y^2}$$ I suspect that the limit does exist as the combined power of $x$ and $y$ is higher in the numerator than in the ...
4
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1answer
453 views

Polar coordinates, line integrals, and the Beltrami Identity

Imagine you are walking along the xy-plane. There is a landmark at the origin of the plane which distorts time at every point on the plane, such that the distortion is a function of the distance ...
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3answers
236 views

Visualizing why a right-angle rotation formula works in polar coordinates

I am trying to get a solid and intuitive handle on polar and spherical coordinates, and I'm getting stuck with what I think should be simple geometry: To find the unit vector in Cartesian coordinates ...
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1answer
71 views

Am I doing this double integral right?

I want to calculate $\iint_R x \ \mathrm{d}A$, where $R$ is the unit disc centered at $(2, 0)$. First, I made the following substitution: $$x' = x-2$$ $$\mathrm{d}x' = \mathrm{d}x$$ $$ ...
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3answers
2k views

Writing a Polar Equation for the Graph of an Implicit Cartesian Equation

If $(x^2+y^2)^3=4x^2y^2,$ then $r=\sin 2\theta$ for some $\theta$. Using $r^2=x^2+y^2, x=r\cos\theta,y=r\sin\theta$, it's easy to get $r^2=\sin^22\theta$. But I don't know what to do next, since ...
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3answers
127 views

Converting a polar coord to the range $0\le\theta\le2\pi?$

I know that you can keep adding/subtracting numbers to a polar coord, but what if I want to be able to take a number and just convert it to its positive equivalent?
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3answers
1k views

Ηοw to find the area of this region

I have two functions $$r=2$$ $$r= 3+2sin\theta$$ and I want to find the area of the yellow region in the picture below. The limits of the integral solving the equation must be ...
2
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1answer
2k views

Horizontal and vertical asymptotes of polar curve $r = \theta/(\pi - \theta) \, , \, \in[0,\pi]$

I as supposed to find the vertical and horizontal asymptotes to the polar curve $$ r = \frac{\theta}{\pi - \theta} \quad \theta \in [0,\pi]$$ The usual method here is to multiply by $\cos$ and ...
0
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0answers
111 views

Set of all points which are a specified angle away from a given point on a sphere.

I have a sphere with a known point on the surface in polar coordinates. I'm looking to find the set of all points which are exactly some angle away from this point in polar form (this should describe ...
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1answer
2k views

Question about the limits of integration using polar coordinates

I haven't been able to find an answer to something I've been thinking about. If you are taking the integral of a circle in polar coordinates you always use the limits for theta as $0$ to $2\pi$. ...
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1answer
171 views

Bijections of the plane

I recently had to deal with polar coordinates and thus wondered: "Polar coordinates" is just a special name for some bijection from $\mathbb{R}^2$ to $\mathbb{R}^2$ that can be very easily visualized ...
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2answers
2k views

how to get $dx\; dy=r\;dr\;d\theta$

In polar coordinate how we can get $dx\;dy=r\;dr\;d\theta$? with these parameters: $r=\sqrt{x^2+y^2}$ $x=r\cos\theta$ $y=r\sin\theta$ Tanks.
0
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1answer
643 views

definition of sinusoidal curve

I have question related with these two definition: In geometry, the sinusoidal spirals are a family of curves defined by the equation in polar coordinates $$r^n = a^n \cos(n \theta)$$ where $a$ is ...
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1answer
254 views

Questions about Hyperbolic Isometries: The Standard Inversion

I have two questions regarding the inversion across the unit circle in the hyperbolic plane. Recall that the hyperbolic plane is a metric space consisting of the open half-plane $$\mathbb{H}^2 = ...
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3answers
4k views

Polar to Parametric Equation?

I'm struggling with this problem, I'm still only on part (a). I tried X=rcos(theta) Y=rsin(theta) but I don't think I'm doing it right. Curve C has polar equation ...
2
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1answer
326 views

Equation for the sensitivity pattern of a bi-directional microphone?

Can anyone give me an equation that expresses the sensitivity pattern of a bi-directional microphone, as a function of azimuth and elevation angle? A bi-directional microphone pattern looks something ...
5
votes
2answers
593 views

How do I write the 2D Dirac delta in a manifestly rotationally invariant form?

Consider the following integral over a 2D plane, $$\iint \mathrm{d}^2\mathbf{k}\ e^{i\mathbf{k}\cdot\mathbf{r}} = 4\pi^2\delta^2(\mathbf{r})$$ This is a Fourier transform of a distribution which is ...
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1answer
492 views

3D parametric equations with polar coordinates

I'm currently studying for my calc 2 midterm and came across this and it completely lost me. I'm not even completely sure where to begin with it. Any ideas? Put $\langle x[r,t],y[r,t],z[r,t] \rangle ...
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1answer
254 views

How to use polar coordinate to represent a $1 \times 1$ square rotated $45^{\circ}$ and translate to $(7,4)$?

How to use polar coordinate to represent a $1 \times 1$ square rotated $45^\circ$ and translated to $(7,4)$? Does the $r(\theta)$ have discontinuous (such at jump from $+5$ to $-2$)? Please help. ...
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1answer
320 views

How to determine a shape is convex by giving polar form polynomial equation?

It is easy to determine concave, convex curve in xy coordinate. But I am placing a question that I only have a polar polynomial equation like r(ang) = a4*ang^4 + a3*ang^3 + .... + a0; How I can tell ...
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2answers
824 views

Express this curve in the rectangular form

Express the curve $r = 9/(4+\sin \theta)$ in rectangular form. And what is the rectangular form? if I get the expression in rectangular form, how am I able to convert it back to polar ...
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6answers
1k views

Why, conceptually, do limaçons $r=a+b\cos\theta$ have dimples when $|\frac{a}{b}|<2$?

Using calculus, I can justify that limaçons—the polar graphs of $r=a+b\cos\theta$ for various nonzero real values of $a$ and $b$—are dimpled when $|\frac{a}{b}|<2$, but that doesn't seem to yield ...
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2answers
126 views

Match the following polar equation to the best description: r^2 = 39 / sin(2θ)

Now, I've guessed the answer, it's a hyperbola, and I know what a hyperbolic function looks like, but I'm having a hard time getting it there. Here's my work so far: First off, $r^2 = x^2 + y^2$ ...
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2answers
8k views

Find the area of the region inside: $r= 6\sin(\theta)$ but outside of $r = 1$

How do we find the area of the region inside $r = 6 \sin(\theta)$, but outside $r = 1$? So, here's my work thus far: First off, we know: $r^2 = x^2 + y^2$ and $\mathrm{sin}(\theta) = y/r$ ...
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3answers
696 views

A square root of i with negative imaginary part

In an ODE class, one assignment question says find the “rectangular” expression z = a + bi (with a and b real) and the “polar” expression |z|, Arg(z) where z is "a square root of i with negative ...
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2answers
8k views

Why is $dy dx = r dr d \theta$ [duplicate]

Possible Duplicate: Explain $\iint \mathrm dx\mathrm dy = \iint r \mathrm d\alpha\mathrm dr$ I'm reading the proof of Gaussian integration. When we change to polar coordinates, why do we ...
3
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1answer
2k views

polar coordinates of Gaussian Distribution with non zero mean

I found that the polar coordinates of 2-dimensional Gaussian distribution with mean zero $$\frac{1}{2\pi\sigma^2}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\exp\big(-({x^2+y^2})/{2\sigma^2}\big) ...
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1answer
1k views

conversion of 2D Gaussian into polar coordinates

Is it possible to convert the 2D Gaussian function in to polar coordinates? ...
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1answer
329 views

Get polar equation from cartesian equation

I have this equation: $x^4 + y^4 = x^2 + y^2$ and I need to convert it to a polar one... I have tried and the result is $$r = \sqrt{\frac{1}{\cos^4\theta + \sin^4\theta}}$$ Is this ok?
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5answers
2k views

Length of $r=3\sin(\theta)$

I have a general understanding of calculating arc length, but this one's a real curve ball. So, I need to find the exact length of $r=3\sin(θ)$ on $0 ≤ θ ≤ π/3$ So the way I've thought of ...
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2answers
262 views

Trying to plot these points in a polar coordinate system

I started with: inside $r_1=5 \sin(θ)$ and outside $r_2=2+\sin(θ)$ and was told to sketch curve in the same polar coordinate system I first set both equal to $0$ and solved to get $\pi$, $2\pi$, and ...