Gradient operator

From HvWiki

The gradient operator yields the slope of a tangent (hyper)plane of a multivariable function. In electromagnetic theory, the gradient of a static scalar potential will yield the field vector. The gradient is often written as

$\textrm{grad} \Phi\,$

or,

$\nabla\Phi.\,$

In three-dimensional rectangular coordinates:

$\nabla\Phi = \textbf{a}_x \frac{\partial\Phi}{\partial x} + \textbf{a}_y \frac{\partial\Phi}{\partial y} + \textbf{a}_z \frac{\partial\Phi}{\partial z}$

In circular cylindrical coordinates:

$\nabla\Phi = \textbf{a}_r \frac{\partial\Phi}{\partial r} + \textbf{a}_\phi \frac{1}{r}\frac{\partial\Phi}{\partial \phi} + \textbf{a}_z \frac{\partial\Phi}{\partial z}$

In spherical coordinates:

$\nabla\Phi = \textbf{a}_r\frac{\partial\Phi}{\partial r} + \textbf{a}_\theta\frac{1}{r}\frac{\partial\Phi}{\partial \theta} + \textbf{a}_\phi\frac{1}{r\sin\theta}\frac{\partial\Phi}{\partial \phi}$

In orthogonal curvilinear coordinates:

$\nabla\Phi = \textbf{a}_1\frac{1}{h_1}\frac{\partial\Phi}{\partial q_1} + \textbf{a}_2\frac{1}{h_2}\frac{\partial\Phi}{\partial q_2} + \textbf{a}_3\frac{1}{h_3}\frac{\partial\Phi}{\partial q_3}$