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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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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57 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
56 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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16 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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95 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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1answer
144 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
134 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
39 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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54 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 ...
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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 ...
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1answer
22 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
38 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 ...
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2answers
45 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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37 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 $xz$ ...
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2answers
36 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 ...
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2answers
43 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
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1answer
29 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 ...
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1answer
16 views

Graphing in Polar Coordinates

I´m currently using polar coordinates to calculate some double and triple integrals. However, I have an small doubt; when you are want to express, lets say, a circle of radius $a$ centered in $(a,0)$ ...
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2answers
50 views

System of equations, limit points

This is a worked out example in my book, but I am having a little trouble understanding it: Consider the system of equations: $$x'=y+x(1-x^2-y^2)$$ $$y'=-x+y(1-x^2-y^2)$$ The orbits and limit sets ...
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1answer
37 views

Conversion of polar coordinate differential 1-forms to xy-plane

I am new to differential geometry (and StackExchange!) and am having some trouble with the conversion of the polar differential one-forms: $dr$ and $d \theta $. How do I express these in terms of ...
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1answer
50 views

Integral by polar coordinates

How to calculate the integral $$\int_0^6\int_0^y x\;dx dy$$ using polar coordinates?$$$$I know that $x=R\cos \theta$ and $y=R\sin\theta$ and that the Jacobian is $R$.
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20 views

Specific cartesian coordinates of an ellipse

I want to do the following: 1.) Ask user for the vertical and horizontal distances of the ellipse 2.) With this information calculate the circumference 3.) Divide the circumference by the closest ...
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2answers
55 views

Better substitution calculating integral?

I'm calculating $$ \iint\limits_S \, \left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}} \right)^\frac{1}{2} \, dA$$ with $$S =\left\{ (x, \, y) \in \mathbb{R}^2 : ...
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2answers
364 views

converting improper double integrals to polar form: what do I do with infinity limits

I need to convert $$ \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}-e^{\frac{x^2+y^2}{5}}dA$$ To polar form. I know $x^2+y^2 = r^2, $ and $dA = rdrd\theta$ But what do I do with the $\infty$ ...
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36 views

Need help converting $z = \ln(x^2 + y^2)$ to polar

The full question is this: Volume of a solid in any region R is given by: $$\int\!\!\!\int_Rf(x,y)dydx $$ where, $$f(x,y) = z = \ln(x^2+y^2)$$ and, $$x^2+y^2=r^2$$ There for, $$dydx = ...
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0answers
49 views

What function has a 3D graph that will look like a spiral into a singularity?

I am trying to draw text spiraling into a black hole, from a more interesting slightly off-orthogonal viewpoint. I think a function that defines a black hole/singularity surface might look something ...
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21 views

Double Integral Mistake with Parametric Equation

I'm trying to figure out the mass of an object bounded by $y=0$ and $y=\sqrt{1-x^2}$ the density at a given point is proportional to its distance from the origin; $\rho(x,y) = kxy$. So I set it up ...
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40 views

Convert geodetic coordinates to cartesian coordinates

I am working on some simulation software that will represent a number of entities in a defined geographic area in the world. The part of the software that I am currently working on is to implement ...
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2answers
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solve polar coordinate integral

Evaluate $$\int_0^R\int_0^\sqrt{R^2-x^2} e^{-(x^2+y^2)} \,dy\,dx$$ using polar coordinates. My answer is $-\frac{1}{2}R(e^{-R^2+x^2}-1)$ but I want to confirm if that's correct And also, when I ...
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0answers
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Find arc centerpoint(x,y) with start(x,y) and end(x,y) in a conical helix

Im trying to script drawing of a conical helix in 3D software, and are stuck at the last arc when its not a full 180 degree arc. I know(calculate) the arc startpoint and endpoint, but how do I find ...
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1answer
26 views

Cartesian to Polar coordinates where alpha is real parameter

I want to convert the following equation in Cartesian form to Polar: $$-y(1+\alpha+x)+x(1-x^2-y^2)$$ so $x = r\cos(\theta)$ and $y = r\sin(\theta)$ I can get this far: ...
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1answer
27 views

Is it possible to write all of the functions in terms of polar form?

Is it possible to write all functions in terms of polar form? For example, the equation of the circle with radius one can be written like $r=1$ I'm wondering whether reform the equations of all curves ...
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62 views

Finding the Equation of a Tangent Line to a Polar Curve

Find the equation of the tangent line to the polar curve: $r=3-3\sin\theta$ at $\theta=\frac{3\pi}{4}$ I have the equation: $$\frac{dy}{dx} ...
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1answer
19 views

Tangent Line of Polar Curve

i start by changing polar coords into x and y and then find the derivatives to get the slope. $$x=(3-3\sin\theta)\cos\theta $$ $$x=3\cos\theta -3\cos\theta \sin\theta $$ and took $x'=(-3\sin\theta ...
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1answer
111 views

Transform second order partial derivatives into polar coordinates

I have the following question: Let $u(x, y)$ be a function with continuous second order partial derivatives. Use the chain rule to transform the expression: $$ x^2\frac{\partial^2u}{\partial ...
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1answer
67 views

Changing to polar form for Green's Theorem

In my text given the integral $\int_{\partial{D}} xy\,dx$, and that $$\int_{\partial{D}} xy\,dx=-\int\int_{D}x\,dx\,dy = - \int\int r\cos \theta\,r\,dr\,d\theta$$ I'm not really understanding the ...
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4answers
59 views

Limit using polar coordinates?

$$\lim_{(x,y) \to (0,0)} \frac{x y^2}{(3x^2 + 4x^2)}$$ How would one calculate above "using polar coordinates"? It was mentioned during class shortly, but we won't be introduced to this until next ...
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1answer
212 views

How to find the limits of integration to get the area for a loop of a lemniscate?

I know how to integrate the squared radius to get the equation that'll give me the area, like such for a lemniscate with $r^2=8\sin(2\theta)$ : $$1/2\int 8sin(2\theta) = 4 \int \sin(2\theta) = 4 * ...
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2answers
39 views

Divergence of vector in spherical coordinates

How should I calculate the divergence for $$\vec{V}=\frac {\vec{r}}{r^2}$$ Is it possible to convert it from spherical coordinates to cartesian?
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1answer
49 views

Function in Polar Coordinates

Let $f,g:I\to\mathbb{R}$ be two function in $C^{k}(I)$, with the property that $f^2(t)+g^2(t)=1, \ \forall\ t\in I$. Is there a function $\theta: I\to\mathbb{R}$, $\theta\in C^{k}(I)$, such that: ...
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1answer
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What is the cartesian equation of $r = 4 + \frac{\sin(\theta)}{2}?$

This is extremely similar to this question, but as there is no r next to the constant 1, when I multiply everything by r I'm going to end up with: $r^2 = 4r + r\frac{\sin(\theta)}{2}$ And I don't ...
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1answer
20 views

Double integral in cylindrical coordinates

I'm having trouble with a double integral problem in cylindrical coordinates. I'm sure the answer is staring me in the face, but I'm missing something. In the multivariable version of the Community ...
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1answer
56 views

Double Integral to Polar Coordinates

Evaluate $$\int_{0}^{2}\int_{0}^{\sqrt{2x-x^2}} \sqrt{x^2+y^2}dydx$$ by converting to polar coordinates. I sketch the region which is a half circle from $0$ to $2$ on the $x$-axis and $0$ to $1$ ...
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1answer
23 views

Graphing A Polar Equation

So, I encountered a question r = -|sinø|. So, I thought the polar graph would look like (2) but it actually looks like (3) and I don't understand why. Can someone explain it to me? I've attached a ...
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2answers
127 views

In polar coordinates, can r be negative?

I'm getting different answers for this. Many websites say that when you get a negative value of r, you flip the coordinate 180 degrees across the pole. However my teacher says that you cannot have a ...
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1answer
29 views

Arc length of a polar curve in terms of theta

Is there an equation for such? I know that there is an equation for such in terms of r, but I must calculate the length of $tan(\theta) = 3/5$ (cartesian equivalent: $y = 3/5x$) from r = 0 to 1.457. ...
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2answers
20 views

Points of intersection for two polar equations question

Why is it that when I try to find the points of intersection for $r=2$ and $r=4*\cos(2\theta)$, I only get the $\theta$ where the reference angle is $\pi/6$? There is clearly another solution between ...
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2answers
47 views

How can I solve these two tough integrals?

\begin{equation*} J_{1} = \int_{0}^{\sqrt{{\pi}/{6}}} \int_{y}^{\sqrt{{\pi}/{6}}} \cos{(x^2)}\,dx\,dy \end{equation*} \begin{equation*} J_{2} = \int\int_{E}\int z e^{(x^2+y^2)} + xe^{x^8}\,dV, ...
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3answers
229 views

Polar coordinates confusion

This seems to be very easy, however I cannot understand, where I am mistaking. Here's the integral to be computed: $$\iint_Dx^2+y^2dydx$$ with $D:=\left\{(x,y)\in \mathbb{R}^2:x \ge0, \; ...
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2answers
28 views

Transforming a cartesian equation to a polar one when it has different x and y denominators?

$$\frac{x^2}{9}+\frac{y^2}{16}=1$$ Needs to be replaced with an equivalent polar equation. I'm sure the identity I'll have to use will be $$x^2+y^2=r^2$$ though other options are: $x=rcos\theta$ ...