Differential Operators in Three Dimensions
Since these operations are common in application, all vector components in this section are presented with respect to the normalized basis.
-
Operator Expression Gradient of a scalar field 
Divergence of a vector field ![\nabla \cdot \mathbf F =
\frac{1}{h_1 h_2 h_3}
\left[
\frac{\partial}{\partial q^1} \left( F_1 h_2 h_3 \right) +
\frac{\partial}{\partial q^2} \left( F_2 h_3 h_1 \right) +
\frac{\partial}{\partial q^3} \left( F_3 h_1 h_2 \right)
\right]](http://upload.wikimedia.org/math/0/8/d/08da56abb22558947b8a754d4cbfc1ab.png)
Curl of a vector field ![\begin{align}
\nabla \times \mathbf F & =
\frac{\hat{ \mathbf e}_1}{h_2 h_3}
\left[
\frac{\partial}{\partial q^2} \left( h_3 F_3 \right) -
\frac{\partial}{\partial q^3} \left( h_2 F_2 \right)
\right] +
\frac{\hat{ \mathbf e}_2}{h_3 h_1}
\left[
\frac{\partial}{\partial q^3} \left( h_1 F_1 \right) -
\frac{\partial}{\partial q^1} \left( h_3 F_3 \right)
\right] \\
& + \frac{\hat{ \mathbf e}_3}{h_1 h_2}
\left[
\frac{\partial}{\partial q^1} \left( h_2 F_2 \right) -
\frac{\partial}{\partial q^2} \left( h_1 F_1 \right)
\right]
=\frac{1}{h_1 h_2 h_3}
\begin{vmatrix}
h_1\hat{\mathbf{e}}_1 & h_2\hat{\mathbf{e}}_2 & h_3\hat{\mathbf{e}}_3 \\
\dfrac{\partial}{\partial q^1} & \dfrac{\partial}{\partial q^2} & \dfrac{\partial}{\partial q^3} \\
h_1 F_1 & h_2 F_2 & h_3 F_3
\end{vmatrix}
\end{align}](http://upload.wikimedia.org/math/2/6/9/2695691b66ce76dbc10aefd856fe5fe8.png)
Laplacian of a scalar field ![\nabla^2 \phi = \frac{1}{h_1 h_2 h_3}
\left[
\frac{\partial}{\partial q^1} \left( \frac{h_2 h_3}{h_1} \frac{\partial \phi}{\partial q^1} \right) +
\frac{\partial}{\partial q^2} \left( \frac{h_3 h_1}{h_2} \frac{\partial \phi}{\partial q^2} \right) +
\frac{\partial}{\partial q^3} \left( \frac{h_1 h_2}{h_3} \frac{\partial \phi}{\partial q^3} \right)
\right]](http://upload.wikimedia.org/math/6/2/c/62cfb877f38541534a7c89c39962c72d.png)
Read more about this topic: Orthogonal Coordinates
Famous quotes containing the words differential and/or dimensions:
“But how is one to make a scientist understand that there is something unalterably deranged about differential calculus, quantum theory, or the obscene and so inanely liturgical ordeals of the precession of the equinoxes.”
—Antonin Artaud (1896–1948)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)