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
19 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
35 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: $$\frac{x^2}{4}+\frac{y^2}{16}=3$$ ...
2
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2answers
64 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 ...
2
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2answers
32 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
36 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
18 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 ...
2
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1answer
30 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
19 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
21 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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0answers
42 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
34 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 ...
2
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1answer
34 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. ...
2
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2answers
53 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 $- ...
1
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1answer
23 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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0answers
38 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
39 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
14 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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0answers
11 views

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

use the equations r=sqrt(x^2 +y^2) and theta=arctan(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 respect to x and y. I have ...
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1answer
26 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
30 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
30 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
24 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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0answers
10 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 ...
2
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1answer
54 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
21 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
18 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
41 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 ...
2
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2answers
60 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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1answer
24 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 ...
1
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1answer
26 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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0answers
47 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
49 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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0answers
9 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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0answers
67 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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5answers
56 views

Find the center and radius of polar circle equation [closed]

Find the radius and center of the circle $$r=2\cos \theta+3\sin \theta$$
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0answers
22 views

why this formula doesn't work?

L=integral from theta(start) to theta (end) r d(theta )......to find the arc length we usually use L= integral from theta(start) to theta (end) sqrt r^2+(dr/dtheta)^2 dtheta why the first formula ...
0
votes
1answer
48 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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2answers
37 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
18 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) ...
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2answers
48 views

A triangle having coordinates $(a\cos\phi, a \sin\phi) , (a\cos\theta, a\sin\theta) , (a\cos\psi, a \sin\psi)$…

A triangle having coordinates $(a\cos\phi, a \sin\phi) , (a\cos\theta, a\sin\theta) , (a\cos\psi, a \sin\psi)$ having its area $$ \Delta = 2a^2 \sin\frac{\theta - \phi}{2}\sin\frac{\phi ...
1
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0answers
15 views

Alpha and Omega Limit Sets in Polar Coordinates [duplicate]

I guess here I am not sure how to get started, I know the definitions: The $ω$-limit sets of points are the set of points that the system of equation approach as time goes to infinity, and the ...
2
votes
1answer
21 views

Use triple to verify that a paraboloid divides a solid in two regions of the same volume, where am I wrong?

Let $S$ be the region over the $xy$ plane and inside the intersection of the cylinder $x^2+y^2=a^2$ and the plane $z=a^2$. I want to verify that the paraboloid $z=x^2+y^2$ divides $S$ into two ...
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2answers
30 views

Polar coordinates, Differentiation

Can someone clarify this step for me please, "The polar coordinate r satisfies $r^2=x^2+y^2$, so by differentiating with respect to t we get $r\cdot\dot r=x\cdot\dot x+y\cdot\dot y$" I am totally ...
-1
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1answer
34 views

Change of Variables Diff Equation Polar [closed]

(a) Show that the general solution to the differential equation $\frac{dy}{dx}=\frac{x+ay}{ax-y}$ can be written in polar form as $r=k{e^a}^\theta$ . b) for the particular case when a = 1/2 , ...
2
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2answers
34 views

Understanding the Jacobian

I was given this problem: Use double integrals to find the area under the curve defined by $r=1+\sin\theta$. We can see that $0\leq\theta\leq2\pi,$ and $0\leq r\leq 1+\sin\theta.$ My question is, ...
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0answers
32 views

Find the volume of the solid described by $x=x^2+y^2$ and the plane $z=y+2$

I'm trying to use triple integrals to find the volume of the solid described by $x=x^2+y^2$ and the plane $z=y+2$. I already determined that the projections of this solid in the plane $xy$ and $xz$ ...
0
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2answers
31 views

Double integral with polar?

I have the following integral : $$\iint\limits_R \operatorname e^{-\frac{x^2+y^2}{2}} \operatorname d\!y \operatorname d\!x $$ Where R is: $$R=\{(x,y):x^2+y^2 \leq 1\}$$ I think I should convert to ...
1
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2answers
41 views

Solve double integral

$$ \int_0^2 \int_0^{4-x^2} \frac{xe^{2y}}{4-y} \, dy\, dx $$ I'm stuck with this problem. I think I should change it so I integrate with respect to $dx \, dy$ but I'm not sure. Any help? Thanks
0
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1answer
27 views

Solve the double integral

I am calculating: $$ \int\int_R (2ax-x^2-y^2)^{\frac{1}{2}} \, dA$$ Where $R$ is the region determined by the inside of $x^2+y^2-2ax=0$ So far, I tried using polar coordinates, wich turns the ...