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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How to numerically determine whether a line on a polar coordinate graph is dominant over another line?

Given 2 or more line segments appearing on a 2-dimensional polar coordinate graph across multiple quadrants having the same range of x-axis endpoints, how do you numerically determine the dominance of ...
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1answer
12 views

Find all polar coordinates of point $P$ where $P = (7, \pi/3)$.

I don't know where to go from here. Answer choices are: a) $(7, \pi/3 + 2n\pi)$ or $(-7, \pi/3 + 2n\pi)$ b) $(7, \pi/3 + 2n\pi)$ or $(-7, \pi/3 + (2n + 1)\,\pi)$ c) $(7, \pi/3 + (2n + 1)\,\pi)$ ...
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1answer
45 views

Physical Proof of Euler's Formula

I would like to construct a geometrical or physical proof of Euler's Formula $e^{ix}=\cos x +i\sin x $. If anyone has constructed such a proof before I would love to see it, if not, I would like some ...
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0answers
19 views

Coordinates in file isn't in the range -180 to 180 respective -90 to 90?

Hello! I'm creating an android version of a PC program (I've contacted the complany who owns the PC program, so it's legal). The program is in the core a GPS, but is used to navigate pre-defined ...
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4answers
32 views

Proving arg(z/w)=arg(z)-arg(w)

I need to prove that $$arg\left(\frac{z}{w}\right)=arg(z)-arg(w)$$ However, I am a little stuck as to how to go about this. I know the proof for $arg(zw)=arg(z)+arg(w)$ happens by letting ...
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35 views

Integration of $e^{-(3x^2+2\sqrt{2}xy+3y^2)}$ over the plane [duplicate]

How to solve the double integral: $$ \Large \int_{-\infty}^\infty\int_{-\infty}^\infty e^{-(3x^2+2\sqrt{2}xy+3y^2)}\,dx\,dy $$ Is it better to transform to polar coordinates?
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1answer
47 views

$\int^\infty_{-\infty}\int^\infty_{-\infty} e^{-(3x^2+2\sqrt 2xy+3y^2)} dx \,dy$ [on hold]

Evaluate $$\int^\infty_{-\infty}\int^\infty_{-\infty} e^{-(3x^2+2\sqrt 2xy+3y^2)} dx \, dy$$ Please give some hints how to proceed.
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1answer
36 views

How do I find the area shared by the circles $r = 2\cos(\theta)$ and $r = 1$?

I figured out the intersection points: $r=2\cos(\theta)$, $r=1$ $2\cos(\theta) = 1$ $\cos(\theta) = \frac{1}{2}$ $\arccos(1/2) = π/3$ (I), $5π/3$ (IV)
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1answer
54 views

Introducing $\mathrm π$ and polar coordinates in real analysis

From time to time, I think about how material from introductory courses like real analysis or linear algebra can be structured in a way I would have liked to see in my freshman days. So recently, I ...
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2answers
37 views

Convert the Polar Equation to Cartesian Coordinates

$$ r^2=\sec 4\theta $$ I graphed this equations using Wolfram Alpha and found it to be 2 hyperbolas. I'm having difficulty showing this using the standard equations $$ x=r\cos\theta \;, \; ...
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1answer
43 views

Finding the area enclosed by 4 functions using polar coordinates

I need to find the area enclosed by $x^2+y^2$ = 4x, $x^2+y^2$ = 2x, y=x and y=0. How do I use polar coordinates here? It seems to me that representing those functions using polar coordinates is too ...
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1answer
48 views

Cannot find link between trigonometric statements and reduced form

I have been trying to find a way to reduce following trigonometric statements to the reduced form below, but without succes. I haven't been able to grasp the typical train of thought I presume I would ...
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1answer
30 views

Polar coordinate double integral

I have to integrate the following integral: $$ \iint \limits_A sin({x_1}^2 + {x_2}^2) dx_1dx_2 $$ over the set: $A=\{x \in \mathbb{R}^2: 1 \leq {x_1}^2 + {x_2}^2 \leq 9,x_1 \geq -x_2\}$ I ...
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1answer
12 views

Find the area enclosed by curve with polar coordinates

I am having a little difficulty finding the area enclosed by the curve, $r(\theta) = 4 + sin\theta + cos\theta$ with $0 \le \theta \le 2\pi$. I tried integrating over $0 \le \theta \le 2\pi$ and $0 ...
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3answers
39 views

Polar Integration of $ r = 2\cos(\theta)$

$ r = 2\cos(\theta)$ has the graph I want to know why the following integral to find area does not work $$\int_0^{2 \pi } \frac{1}{2} (2 \cos (\theta ))^2 \, d\theta$$ whereas this one does: ...
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0answers
14 views

Polar coordinates used to evaluate a function containing a branch cut

I'm having a lot of trouble understanding how to approach these kinds of problems, if anyone could explain the approach, it would be really helpful. The problem is as follows: The function $f(z)$ is ...
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1answer
62 views

Two-dimensional limit, is my approach correct?

The limit is $$\lim_{(x,y)\to(0,0)}\frac{x^3y}{x^4+y^2}$$ As usual, I tried checking along particular paths, namely the axes and the curves $y=mx^n$ for various values of $n$, but to no avail; all the ...
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0answers
19 views

From what source should I learn about analytic functions given in polar coordinates?

In the Calculus 1 course that I am currently taking, we only discussed functions given in polar coordinates as some sort of side note, but I am eager to explore them more thoroughly. Namely, what I am ...
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1answer
34 views

Polar coordinate system : Is radial coordinate is a function of angular coordinate?

In polar coordinate system: The polar coordinates $r$ is called the radial coordinate and $\theta$ is called the angular coordinate, often called the polar angle. I am confused when answering the ...
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2answers
59 views

Write ODE in Polar Coordinates [closed]

I want to write this ODE system in polar coordinates (r,$\theta$). $$\dot x =x-y-x^3 $$ $$\dot y = x+y-y^3$$
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3answers
52 views

Real and imaginary part of $ (1-i\sqrt{3})^6$

i am a bit stuck here. As the title says i try to find out how to write complex numbers like for example$$ (1-i\sqrt{3})^6$$ in the normal representation$$ z = x + i*y$$ I already found out that the ...
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1answer
30 views

When looking at motion in a circle, why do they say that $ r \dot{\theta}$ is transverse velocity when it doesn't look like it is a vector?

In my lecture notes it says that $r \dot{\theta}$ is called the transverse velocity of a particle if it is travelling in a circle. What I don't understand is why this is called a velocity when neither ...
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2answers
32 views

Two variable limit

Suppose I have a function which is defined in different parts, for example: $$f(x,y)=y\cos\left(\frac{x}{y}\right)\ \ \ y\neq0$$ $$f(x,0)=0$$ and I have to calculate the limit when $(x,y)\rightarrow ...
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0answers
19 views

Inversion of Rose Curve in Unit Circle

The inversion of a polar curve r(t) in the unit circle is given by 1/r(t). A rose is a polar curve defined (eup to similarity) by an equation of the form: r(t) = cos(nt) or r(t) = cos(p/q t) Does ...
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2answers
53 views

Why does the radius come before the angle?

Based on my understanding, when delineating two variables (for a coordinate system or otherwise) convention is to label the 'independent variable' first, then the 'dependent variable'. So for a ...
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0answers
32 views

Find the flux through a closed volume with the divergence theorem and using the definition

Given the vector field F(x,y,z)=(xy,xy,z) and $D= \{(x,y,z) \in R^3 : x^2 + y^2 + z^2 \le 4, x^2 + y^2 \le 1, z\ge 0 \}$ Find the flux through ...
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1answer
45 views

arc length of the polar curve $r^2= \sin2\theta$

given curve is $r^2 = \sin2\theta $ I got $L= \int_0^{2\pi} \sqrt{r^2+ ({\dfrac{dr}{d\theta}})^2}\ d\theta$ = $\int_0^{2\pi} \sqrt{\dfrac{1}{r^2}} d\theta = \int_0^{2\pi} ...
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1answer
27 views

Area of the region inside $r = 1 - \cos(\theta)$ and also inside $r = \cos(\theta)$

Pretty simple polar integration question that I've been having trouble with... The question says it all. I identified the limits of integration by setting $1 - \cos(\theta) = \cos(\theta)$ so that ...
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2answers
37 views

Integrating exponential function with elliptic bounds

I am trying to integrate the following: $$\iint_R\exp\left(\frac{x^2}{4}+\frac{y^2}{16}\right)\:\mathrm{d}A$$ With the region $R$ having the bounds: $$\frac{x^2}{4}+\frac{y^2}{16}=3$$ ...
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2answers
72 views

Does the inverse function theorem fail for $\frac {\partial r}{\partial x}$

This is a follow-up to a question that I answered (though, not well enough). Why is it that $\frac {\partial r}{\partial x} = \cos(\theta) = \frac {\partial x}{\partial r} = \frac {\partial}{\partial ...
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2answers
36 views

Suppose that two polar curves are given by: $R_1 = \cos(2\theta)$ and $R_2 = \sin(3\theta)$. Find the smallest positive solution exactly.

Suppose that two polar curves are given by: $R_1 = \cos(2\theta)$ and $R_2 = \sin(3\theta)$. Find the smallest positive solution exactly. I know that we are looking for the smallest positive value ...
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4answers
37 views

Suppose $x = 3 - 2i$ and $y = 4 + i$. Find both square roots of y. Then indicate which one is the principle square root.

Suppose $x = 3 - 2i$ and $y = 4 + i$. Find both square roots of $y$. Then indicate which one is the principle square root. Use the polar form of complex numbers to accomplish this task. I'm not ...
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0answers
21 views

Polar coordinate for complicated curves

In general polar representation of a closed curve is done by coordinate $(\theta,r(\theta))$, $\theta\in (0,360)$. When working with real data, I got a closed curves whose graph looks like the below ...
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1answer
32 views

Convert $\frac{1+ \sqrt{3i} }{1- \sqrt{3i} }$ to polar form

How do I convert $\frac{1+ \sqrt{3i} }{1- \sqrt{3i} }$ to polar form? I came across it in this question but I don't know much about complex numbers and have no idea how to figure out $\theta$.
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1answer
21 views

Domain of a Bounded Archimedian Spiral???

So I have a question about a bounded Archimedian Spiral: In one context I get that an Archimedian Spiral's domain and range are all Reals. Thus if I'm looking at what appears to be a bounded spiral: ...
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2answers
23 views

Orthonormal basis in a cylindrical coordinate system

So I am supposed to show if these vectors make an orthonormal basis in a cylindrical coordinate system. $\vec e_p=\bigl(\begin{smallmatrix} cos(\theta )\\ sin(\theta )\\0 \end{smallmatrix}\bigr); ...
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43 views

How to represent $y = ax^2 + bx$ using polar coordinate system?

How to represent $y = ax^2 + bx$ using polar coordinate system ? I want to find the length of the curve by polar coordinate system. I've tried to $x\mapsto r\cos \theta$, $y\mapsto r\sin \theta$. ...
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1answer
41 views

Find the area of the region that is enclosed by the cardioid $r=2+2\sin(\theta)$

We just learned polar integration, so I know that's how we're supposed to do it. I have a problem though: I'm getting a negative answer. What I did: Using the graph, which is: I figured out that ...
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1answer
39 views

Find the Area Using Polar Coordinates and a Double Integral

Of the area inside the smaller loop of the equation $r = 1-2sin\theta$ Here's my attempt at a solution: The shape has an inner and an outer loop, both of which will terminate at the origin. ...
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2answers
58 views

Inconsistent answers when implicitly differentiating polar identities

Currently doing a problem where I need to find $\frac {\partial \theta}{\partial x}$. However, for $\tan(\theta)= \dfrac yx$, $\frac {\partial\theta}{\partial x}$ is yielding $- ...
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1answer
24 views

I can't figure out how to solve the polar integral for finding the area!

I have: $$ \int_{}^{} \int_{}^{}r\,drd\theta.$$ And I have to find the area bounded by $r=2(2-\sin(\theta))^{1/2}$. I understand how to find the limits of integration for dr, but how would I find ...
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39 views

Why am I evaluating this polar integral wrong?

I have: $$ \int_{0}^{6} \int_{0}^{y}xdydx.$$ I drew a picture already which is just a triangle in the first quadrant. I then changed the cartesian coordinates into polar coordinates, which came out ...
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1answer
41 views

How do you find the limits of integration without drawing a picture?

Consider the integral $$ \int_{-1}^{1} \int_{0}^{\sqrt{1-x^2}}dydx.$$ I need some help understanding how to find the new limits of integration if I'm evaluating the integral in polar coordinates. ...
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0answers
15 views

Domain of a Bounded Polar Archimedian Spiral???

So I have a question about a bounded Archimedian Spiral: In one context I get that an Archimedian Spiral's domain and range are all Reals. Thus if I'm looking at what appears to be a bounded spiral: ...
0
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1answer
16 views

Polar coordinates doubt (Graph of $r \le 1$)

I have a doubt. I have to plot the graph of $r \le 1$. Now, according to me, it should be a circular disc with center origin and radius 1 unit. But, some of my friends say that it should be the whole ...
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1answer
45 views

Use the chain rule to convert the Laplace equation in (x,y) coordinates into an equivilent differental equation in (r,theta) coordinates. [duplicate]

use the equations $r=\sqrt{x^2 +y^2}$ and $\theta=\arctan(\frac{y}{x})$. I was able to get the partial derivative of of $r$ with respect to $x$ and $y$ and the partial derivative of $\theta$ with ...
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1answer
27 views

Domain in polar coordinates

I have this domain $A=\{ (x,y) \in R^2 : x^2+y^2 \ge4, x^2+y^2-2x-2y\le0 \}$ It's right the change in polar coordinates : $$\{ (r,\theta): \theta \in [0,\frac{\pi}{2}], r \in ...
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1answer
36 views

Write in polar form

I've been giving the the following: $$z = -3e^{-i\pi/5}$$ How do I write that in polar form? I understand that -3 is not correct, since the absolute distance must be $\ge 0$. What do I need to do ...
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0answers
32 views

Complex polar co-ordinates

We know that rectangular co-ordinates $(x, y)$ can be written as a complex number $re^{i\theta}$ where $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1} \big(\frac{y}{x}\big)$ and $r,\theta \in ...
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3answers
27 views

Express angular position of the Earth as a function of time

Say I have for example the Earth orbiting the Sun (in circular orbit) and I want to express angular position (in radians) as a function of time. Would I be correct in thinking that $2\pi/t$ does the ...