Transforms

Coordinate System Transforms

Scalar Transforms

Vector Transforms

For certain magnetic sources, like a sphere or solenoid, their field equations are more conveniently written in non-cartesian coordinates. The resulting vector fields then need to be transformed back into cartesian coordinates of the form \(\mathbf{B} = B_x \mathbf{\hat{x}} + B_y \mathbf{\hat{y}} + B_z \mathbf{\hat{z}}\) or \(\mathbf{B} = B_x \mathbf{\hat{x}} + B_y \mathbf{\hat{y}}\).

Polar to Cartesian

\(\mathbf{B} = B_r \mathbf{\hat{r}} + B_\phi \mathbf{\hat{\phi}}\)

\[ \begin{align} B_x &= B_r \cos \phi - B_\phi \sin \phi \\ B_y &= B_r \sin \phi + B_\phi \cos \phi \end{align} \]

Cylindrical to Cartesian

\(\mathbf{B} = B_r \mathbf{\hat{r}} + B_\phi \mathbf{\hat{\phi}} + B_h \mathbf{\hat{h}}\)

\[ \begin{align} B_x &= B_r \cos \phi - B_\phi \sin \phi \\ B_y &= B_r \sin \phi + B_\phi \cos \phi \\ B_z &= B_h \end{align} \]

For a solenoid or cylinder \(B_\phi = 0\), thus:

\[ \begin{align} B_x &= B_r \cos \phi \\ B_y &= B_r \sin \phi \\ B_z &= B_h \end{align} \]

Spherical to Cartesian

\(\mathbf{B} = B_r \mathbf{\hat{r}} + B_\theta \mathbf{\hat{\theta}} + B_\phi \mathbf{\hat{\phi}}\)

\[ \begin{align} B_x &= B_r \sin\theta \cos\phi + B_\theta \cos\theta\cos\phi - B_\phi \sin\phi \\ B_y &= B_r \sin\theta\sin\phi + B_\theta \cos\theta\sin\phi + B_\phi \cos\phi \\ B_z &= B_r \cos\theta - B_\theta \sin\theta \end{align} \]

Misc

The centroid of a non-self-intersecting closed polygon defined by \(n\) vertices, is the point (\((C_x, C_y)\) where

\[ C_x = \frac{1}{6A}\sum_{i=0}^{n-1}(x_i+x_{i+1})(x_i\ y_{i+1} - x_{i+1}\ y_i) \]

and

\[ C_y = \frac{1}{6A}\sum_{i=0}^{n-1}(y_i+y_{i+1})(x_i\ y_{i+1} - x_{i+1}\ y_i) \]

where \(A\) is the polygon's signed area:

\[ A = \frac{1}{2}\sum_{i=0}^{n-1} (x_i\ y_{i+1} - x_{i+1}\ y_i) \]

Also written as:

\[ A = \frac{1}{2} \left( \begin{vmatrix} x_1 & x_2 \\ y_1 & y_2 \\ \end{vmatrix} + \begin{vmatrix} x_2 & x_3 \\ y_2 & y_3 \\ \end{vmatrix} + \cdots + \begin{vmatrix} x_n & x_1 \\ y_n & y_1 \\ \end{vmatrix} \right) \]

For a convex polygon, if \(A > 0\) the vertices are listed in counter-clockwise order, and clockwise if \(A < 0\).