# Tagged Questions

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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### Finding the polar form of a complex number

I have the following complex numbers : -3,18 +4,19i I can calculate $r=\sqrt{a^2+b^2}$ Which gives r=5,26 now I know that cos $\theta = \frac{a}{r}$ gives $\theta=127,20$ degrees But when I do ...
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### Double integral with Polar coordinates - hard example

Calculate using polar coordinates: $$\iint_{D}^{} (x^2+y^2)^\frac{1}5 \ dx \ dy$$ where D is the region inside the circle with radius 1. Working: D: $\ x^2+y^2=1 \\$ so $0 \leq r \leq 1 \ \ ,$ ...
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### Operators in polar coordinates in n-dimensions

I want help on converting differential operators such as the reduced wave operator (L=Δ+c) and the biharmonic operator (L=Δ^2) from Cartesian to spherical coordinates in n-dimensions. For example I ...
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### Finding the area of a polar region

I am trying to find the area inside the curve $$r = 2 + \sin2\Theta + \cos3\Theta .$$ It's a very weird looking function after graphing, and I'm not quite sure how I'm supposed to proceed. There's ...
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### How to change the equation to polar form?

Compute $\displaystyle\int^\infty_{-\infty} dx\displaystyle\int^\infty_{-\infty} dy\displaystyle\int^\infty_{-\infty} dz \delta\left(\sqrt{x^2 +y^2+z^2} - R\right)$.
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### Parametrization of a rotating surface

What is the parametrization of a surface obtained by rotating the circle $(y − 3)^2 + z^2 = 1, x = 0$ about the z-axis. I came up with the parametrization $S(r,θ) = (r , 3+cosθ , sinθ)$, is it ...
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### How to calculate polar angle of point given a reference point?

I want to calculate polar angle of some points based on different reference points. Usually polar angle is calculated based on reference point (0,0). What is the procedure to calculate polar angle ...
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### Simple proof of the Cauchy-Crofton formula on the sphere?

Let $\gamma$ be a regular curve on the sphere. In a lecture, the following result was used $$L(\gamma)=\frac 14 \int_{S^2} \sharp (\gamma \cap \xi ^\perp)d\xi$$ $\xi^\perp$ is the plane with normal ...
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### Does a plane curve with polar equation $r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$ have a name?

Does a plane curve with polar equation $$r=\lambda_1\cos^2\theta+\lambda_2\sin^2\theta$$ where both $\lambda_i>0$ have a name? It's very similar to hippopede, also known as lemniscate of Booth, ...
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### What are the characteristics of functions that look the same in both polar and rectangular graph?

Today, I am doing practice for SAT. In a textbook example, I see $$r=\frac{1}{\sin\theta}$$ My textbook is telling me that this particular function looks the same whether it's graphed on a polar or ...
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### Double integration in polar coordinates between two circles

I am trying to integrate converting to polar coordinates, between two circles. $$A = \iint_D x \,\mathrm{d}A$$ Ant the domain of integration is set to be the region in the first quadrant between ...
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### Dot product of gradient and tangent vector

Using polar coordinates with variables $r$ and $\theta$. Let $\vec{r}$ be the position vector. Consider $\nabla \theta \cdot \frac{d\vec{r}}{d\theta}$. This is the dot product of the gradient normal ...
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### Polar Coordinates to Cartesian - Finding Y component

I have the following diagram and frame: I am trying to find out what the equation is that matches XYZ to RThetaPhi. Basically, I need an expression that gives Ys in terms of RThetaPhi. My problem ...
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### Difference between Euler form and polar / trig form of a complex number

After some readings, I have found out that the difference between the polar / trigonometric form and the Euler form of a complex number consists on the fact that in the first case is expressed the ...
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### Double integrals in polar coordinates — Multivariable

I've done some research on this topic but I am quite confused about finding the area under a specific volume in polar coordinates. Let's have an example, how would we find the volume of a hyperboloid ...
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### find the area of the region lying inside the circle $r=6$ and inside the cardioid $r=4-3\sin \theta$.

Well, I drew a graph to visualise it and I found the interceptions $\theta=\arcsin \left(-\frac{2}{3}\right)$. From the graph, by symmetry, I found that the area of region from $\theta$ to $\pi/2$ and ...
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### Find the area of the region lying outside a circle r=7 and inside the cardioid r=6+7sin theta

So this is the question I have problem dealing with. I know that firstly I need to equate $7$ and $6 + 7\sin \theta$ to get the intersection. And then I am supposed to apply the formula.. But I am ...
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### Arc Lengths of Points Tangent to a Logarithmic Spiral

Suppose we are given distinct array of $N$ vertices (or Cartesian points) $V_n =(v_1, v_2, ... v_n), v_i \in \mathbb{R}^2$. Taking $v_1$ to be the origin of a logarithmic spiral whose curve ...
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### Predict a point when you are given initial measurements

From given (x,y) sensor measurements, output by a robot, I need to find robot's heading direction and predict the next location. I have an algorithm that when programmed gives me the correct answer, ...
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### Tangent of angle between tan line and radial line

How can I use the fact that if the curve whose polar equation is $r=f(\theta)$ is rotated about the pole through an angle $\phi$, then an equation for the rotated curve is $r=f(\theta-\phi)$ to prove ...
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### Convert $r^2= 9 \cos 2 \theta$ into a Cartesian equation
This is how I tried so far... $r^2= 9 \cos 2 ( \theta)$ $\cos (2 \theta) = \cos ^2 (\theta) - \sin^2 (\theta)$ and $r^2= x^2 + y^2$ so, it will become \$x^2 + y^2 = 9 [\cos^2 (\theta) - \sin^2 ...