MATH 251 Lecture 15

« previous | Wednesday, February 22, 2012 | next »

Derivatives of Polar Coordinates

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathrm{d}r &= r_x \mathrm{d}x + r_y \mathrm{d}y \\ \mathrm{d}\theta &= \theta_x \mathrm{d}x + \theta_y \mathrm{d}y \end{align}}

${\begin{pmatrix}\mathrm {d} x\\\mathrm {d} y\end{pmatrix}}={\begin{pmatrix}\cos {\theta }&-r\sin {\theta }\\\sin {\theta }&r\cos {\theta }\end{pmatrix}}{\begin{pmatrix}\mathrm {d} r\\\mathrm {d} \theta \end{pmatrix}}$

Inverse Matrix

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} A &= \begin{pmatrix} a & b \\ c & d \end{pmatrix} \\ A^{-1} &= \frac{1}{\mathrm{det}{\left(A\right)}} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \end{align}}

Solution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{pmatrix} \cos{\theta} & -r\sin{\theta} \\ \sin{\theta} & r\cos{\theta} \end{pmatrix}^{-1} = \begin{pmatrix} \cos{\theta} & \sin{\theta} \\ -\frac{\sin{\theta}}{r} & \frac{\cos{\theta}}{r} \end{pmatrix}}