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→Vectors in Arbitrary Coordinate Systems: continuing
In other words, the transformation of vector components is just a matrix multiplication by the Jacobian matrix
In other words, the transformation of vector components is just a matrix multiplication by the Jacobian matrix
::<math>J(u, v, w / x, y, z)\,</math>
::<math>\begin{bmatrix}J(u, v, w / x, y, z)\end{bmatrix}</math>
For polar coordinates in two dimensions, we have:
::<math>\begin{bmatrix} A_r \\ A_\theta \end{bmatrix} = \begin{bmatrix}J(r, \theta / x, y)\end{bmatrix} \begin{bmatrix} A_x \\ A_y \end{bmatrix}</math>
or
::<math>\begin{bmatrix} A_r \\ A_\theta \end{bmatrix} = \begin{bmatrix} \frac{\partial r}{\partial x} & \frac{\partial r}{\partial y} \\ \frac{\partial \theta}{\partial x} & \frac{\partial \theta}{\partial y}\end{bmatrix} \begin{bmatrix} A_x \\ A_y \end{bmatrix}</math>
Working this out:
::<math>\begin{bmatrix} A_r \\ A_\theta \end{bmatrix} = \begin{bmatrix} \cos \theta & \sin \theta \\ \frac{- \sin \theta}{r} & \frac{\cos \theta}{r} \end{bmatrix} \begin{bmatrix} A_x \\ A_y \end{bmatrix}</math>
And, going the other way:
::<math>\begin{bmatrix} A_x \\ A_y \end{bmatrix} = \begin{bmatrix} \cos \theta & - r \sin \theta \\ \sin \theta & r \cos \theta \end{bmatrix} \begin{bmatrix} A_r \\ A_\theta \end{bmatrix}</math>
=Introduction to Integrals in Multivariable Calculus=
=Introduction to Integrals in Multivariable Calculus=