Gradient with predefined sampling distance.

\[ \text{grad}(f) = \begin{pmatrix} \partial_x f \\ \partial_y f \end{pmatrix} \]

The fire-and-forget version of gradH.

Divergence with predefined sampling distance. Named to avoid clashing with integer division.

\[ \text{div}(f) = \partial_xf_x + \partial_y f_y \]

The fire-and-forget version of divH.

Curl with predefined sampling distance. Note that in two dimensions, the curl is simply a number. For a different 2D adaptation of the \(\text{curl}\) operator, see curlZ.

\[ \text{curl}(f) = \partial_x f_y - \partial_y f_x \]

The fire-and-forget version of curlH.

Curl of a purely-z-component 3D vector field, which is another common way (than curl) to implement a two-dimensional version of \(\text{curl}\):

\[ \text{curl}_z(\psi) = \left[\text{curl}_{3D} \begin{pmatrix}0\\0\\\psi\end{pmatrix}\right]_{\text{2D part}} = \begin{pmatrix}\partial_y\psi\\-\partial_x\psi\end{pmatrix} \]

Because of this, the result is always divergence-free, because \(\forall f. \text{div}(\text{curl}(f))=0\).

Using curlZ to create divergence-free flow fields is similar to using grad to create curl-free force fields. The argument for curlZ is known as the vector or stream potential in fluid dynamics, with the resulting vector field being known as a stream function.

The fire-and-forget version of curlZH.

Laplacian with predefined sampling distance.

\[ \nabla^2f = \partial^2_x f + \partial^2_y f \]

The fire-and-forget version of laplaceH.

Gradient with customizable sampling distance. The configurable version of grad.

Divergence with customizable sampling distance. The configurable version of divergence.

Curl with customizable sampling distance. The configurable version of curl.

Curl of the z component of a 3D vector field, with customizable sampling distance. The configurable version of curlZ.

Laplacian with customizable sampling distance. The configurable version of laplace.