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
281 views

Find volume of a revolved solid by integrating wedges.

So, lets say that I wanted to find the volume of the solid formed by rotating the area between $f(x)=\sqrt{1-x^2}, 0<x<1$ and the $x$ axis around the $y$ axis. (This example is simply a ...
2
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2answers
145 views

Integral variable substitution using Hausdorff measure

Suppose we have positive density $q$ with "good" qualities (continuity, etc..). I need to calculate this integral: $$\int_B q(\textbf{z}) d \textbf{z},\ \textbf{z} \in \mathbb{R}^d,$$ where $B \subset ...
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3answers
493 views

Expressing $e^z$ where $z=a+bi$ in polar form.

I am reading a passage of text that states: "We can use the fact that $e^{a+bi}=e^a(\cos b+i\sin b)$ has polar form $\left<e^a,b \right>$ to verify that complex exponentials have various ...
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1answer
223 views

Complex Numbers and polar form

I am given the following information: $$x[n]= s^n,\qquad n=0,\pm 1,\pm 2,\ldots$$ where $s=\sigma + j\omega = re^{i\theta}$ is a complex number in general. I was wondering how the following is ...
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1answer
3k views

How to write parametric equations for a given polar equation?

I'm doing an extra credit problem for math, we haven't learned too much on this topic. The instructions are: Write parametric equations for the given polar equation. The problem is: $r = ...
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1answer
1k views

Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation.

Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation. Let $t= \theta$, $r=5$, $x=r\cos t$, $y=r\sin t$. I began with $x=5\cos t-5=5(\cos t-1)$ and $y=5\sin t$. Is that ...
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1answer
347 views

Finding a geodesic on a plane using polar coordinates

This is from my homework on PDE. I need to find a geodesic on a plane using polar coordinates. Now, I know $dl^2 = x^2+y^2$ hence $l=\int \sqrt{dx^2+dy^2}$, but I get stuck while converting ...
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2answers
138 views

Laplacian in polar coordinates

I am stuck with an exercise that requires me to find the Laplacian $\Delta u=(D_x^2u+D_y^2u)$ of a 2d-function $u$ in polar coordinates (in the standard Euclidean plane). I found the following ...
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1answer
141 views

Explain why for $r=1-a \cos^2(3\theta)$ the leaves have the same size only in the case $a=1$ and $a=2$

Please explain why for $r=1-a\cos^2(3\theta)$ the leaves have the same size only in the case $a=1$ and $a=2$. Does anyone have an answer to this please?
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1answer
541 views

Transform integral into polar coordinates

At university we are given a voluntary hand in in the use of maple/matlab, in that regard I have a double integral I am in dire need to compute, using first cartesian then polarcoordinates. ...
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1answer
1k 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 ...
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1answer
64 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
64 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
237 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: ...
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1answer
131 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
214 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
135 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.
2
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1answer
409 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
189 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
478 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
1k 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. ...
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2answers
88 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
70 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
984 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 = ...
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2answers
214 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
463 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
247 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
72 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
141 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 ...
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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
192 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.
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1answer
711 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
281 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
373 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
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2answers
673 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 ...
0
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1answer
525 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
283 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
345 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
883 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 ...