This video helps answer the question: "Are polar coordinates useful?" Turns out they are the backbone to basically every device in the digital era. Without them, we would not have any modern inventions.
- Fourier Transforms - The first 3 minutes in class is sufficient. Keep watching, on your own, if it spiked your interest.
Brain Gains
- We know $x=r\cos\theta$. For the curve $r=2\sin\theta$, find $\ds \frac{dr}{d\theta}$ and $\ds \frac{dx}{d\theta}$.
Solution
We have $\frac{dr}{d\theta} = 2\cos\theta$ and $x=(2\sin\theta)(\cos\theta)$, which means $$\begin{align} \frac{dx}{d\theta}&= (2\sin\theta)'(\cos\theta)+(2\sin\theta)(\cos\theta)'\\ &= (2\cos\theta)(\cos\theta)+(2\sin\theta)(-\sin\theta). \end{align}$$
- For the change of coordinates $x=2u+3v^2$ and $y=4u^3+5v$, write the differential $(dx,dy)$ in the form
$$ \begin{pmatrix}dx\\dy\end{pmatrix}= \begin{pmatrix}?\\?\end{pmatrix}du+ \begin{pmatrix}?\\?\end{pmatrix}dv.$$
Solution
First, note that $$\begin{align} dx&=2du+6vdv\\ dy&=12u^2du+5dv. \end{align}$$ Rewriting this in vector form gives $$ \begin{pmatrix}dx\\dy\end{pmatrix}= \begin{pmatrix}2\\12u^2\end{pmatrix}du+ \begin{pmatrix}6v\\5\end{pmatrix}dv. $$
- For the curve $r=2+2\sin\theta$, graph the curve in the $r\theta$ plane.
Solution
Here is a Desmos Graph.
- For the curve $r=2+2\sin\theta$, graph the curve in the $xy$ plane.
Solution
Here is a Desmos Graph.
Group problems
- Review: for the equation $z=x^2y+3y^2$, show that $\ds\frac{dz}{dt} = 2xy\frac{dx}{dt}+x^2\frac{dy}{dt}+6y\frac{dy}{dt}$.
- Compute the differential $dA$ for the area function $A=xy$.
- We know $x=r\cos\theta$. Explain why $dx = \cos\theta dr-r\sin\theta d\theta$.
- We know $y=r\sin\theta$. Compute $dy$ in terms of $r,\theta,dr,d\theta$.
- Plot the curve $r=3-3\sin\theta$ in the $r\theta$ plane, and then in the $xy$-plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\cos\theta$ easy to compute. Did you get a heart shaped object?]
- Plot the curve $r=3\cos2\theta$ in the $r\theta$ plane, and then in the $xy$-plane. [Hint: Make an $(r,\theta)$ table, but pick values for $\theta$ that make $\sin2\theta$ easy to compute (multiples of 45 degrees). Did you get a clover?]
- Plot the curve $r=4-4\cos\theta$ in both the $r\theta$-plane, and the $xy$-plane.
- Plot the curve $r=3\sin2\theta$ in both the $r\theta$-plane, and the $xy$-plane.
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