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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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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66 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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21 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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53 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
71 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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55 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
38 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
34 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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47 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
58 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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53 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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61 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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509 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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41 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
86 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
42 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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39 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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47 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
37 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
47 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
80 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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45 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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322 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
47 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
62 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
28 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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41 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
55 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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24 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: ...
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1answer
21 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
173 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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34 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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52 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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35 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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31 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 ...
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20 views

Horizontal and Vertical Tangents of Limicons

I feel like I am over thinking this problem, and am probably just confusing myself... So I need to find the values of $t$ where the equation $r=a+b\cos(t)$ has horizontal and vertical tangents, for ...
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130 views

Find the highest point on the cardioid $r = 1 + \cos(\theta)$

I'm stuck on this. I don't know where to start! The problem: Find the highest point on the cardioid $r = 1 + \cos(\theta)$
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1answer
163 views

Perimeter of a region polar curve

I'm having some trouble with this problem: Find the length of the entire perimeter of the region inside $r = 11 \sin(\theta)$ but outside $r = 3$. I am using the formula $$\int_a^b\! \sqrt{r^2 ...
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1answer
120 views

Polar coordinate line to slope intercept form

I'm finding it very difficult to find an answer on google and in my math book on this. The question give to us is: A curve with polar equation $$ r= \frac{39}{9\sin\theta+19\cos\theta} $$ ...
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4answers
70 views

Graph r=6sin(θ)

I'm stuck on this one. I've tried converting it to Cartesian coordinates but I couldn't. I know I could figure it out by testing a bunch of values for θ, but I'd like to know how to do it a better ...
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2answers
68 views

changing $r=1+2r\cos \theta$ to its cartesian equivalent

My textbook says the polar equation, $r=1+2r\cos \theta$, its cartesian equivalent is $y^2-3x^2-4x-1=0.$ I understand that I get this if I square $r$; $r^2=x^2+y^2=(1+2x)^2.$ But don't I need to ...
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27 views

To what scope polar coordinate makes sense?

In basic calculus, one partial-differentiate a differentiable function whose domain is an open set or a closed set etc. However how formally this process works? Here is a reference : definition of ...
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1answer
30 views

Algebraic step on a trig expressiom in linear algebra

$$W = ||V||(\cos(\varphi)\cdot \cos(\theta) - \sin(\varphi)\cdot\sin(\theta), \cos(\varphi)\cdot\sin(\theta) + \sin(\varphi)\cdot\cos(\theta))$$ $$= (v_1 \cos(\theta) - v_2 \sin(\theta), v_1 ...
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63 views

Line integral of a conservative fields over a circle

I need to show that moving the curve to a simply connected region, the integral of the field over the curve will be $0$. Given $F(x, y) = ((-y)/(x^2+y^2 ), (x/(x^2+y^2 ))$, and $γ$ circle of ...
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1answer
63 views

Unable to solve any Euler questions. Fundamental error I cannot find

Good day, I have been trying to solve Euler based questions for a day now. And i realize I still cannot solve a single one, and am getting errors for all my questions. I feel like I am getting ...
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18 views

calculating position of a point knowing two reference lengths

Hi, I would like to know if there is a way to calculate a unique position for Point A knowing the lengths l1 and l2 which are variable string lengths. Point A can move within the range shown below. ...
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108 views

What is a complex number that can't be written in polar form?

What is the cartesian form of a complex number that can't be written in polar form? Why can't it be written in polar form?
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174 views

Show that $u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$ is in $W^{1,n}(U)$, where $U=B(0,1)\subset\mathbb{R}^n$.

The entire problem statement is: Let $n>1$ and let $U=B(0,1)\subset\mathbb{R}^n$. Show that $u:U\to\mathbb{R}$ given by $$u(x)=\ln\left(\ln\left(1+\frac{1}{|x|}\right)\right)$$ is in ...
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244 views

Find Cartesian coordinates of polar curve $r =5\sin(\theta) + 5\cos(\theta)$

Polar equation of the form $r = 5\sin(\theta) + 5\cos(\theta)$ The Cartesian equation is of the form $(x-A)^2+(y-B)^2 = R^2$ Find $A,B$, and $R$. Guess: Let $x = R\cos(\theta) + A$ and $y = ...
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3answers
73 views

How can I calculate angles between objects at the sky?

There is a polar coordinate system which represents the sky from an observer. The elevation angle is 0 to 90 degrees which corresponds to horizon to zenith. The azimuth angle is 0 degrees (north) ...