Gradient and directional derivative in general coordinates $(\psi , \theta , \zeta )$

The gradient of a scalar function $f (\psi, \theta, \zeta)$ is readily calculated from the chain rule,

$$\nabla f = \frac{\partial f}{\partial \psi} \nabla \psi + \frac{\... ...artial \theta} \nabla \theta + \frac{\partial f}{\partial \zeta} \nabla \zeta .$$ (128)

Note that the gradient of a scalar function is in the covariant representation. The inverse form of this expression is obtained by dotting the above equation by the three contravariant basis vectors, respectively,

$$\frac{\partial f}{\partial \psi} = (\mathcal{J} \nabla \theta \times \nabla \zeta) \cdot \nabla f,$$ (129)

$$\frac{\partial f}{\partial \theta} = (\mathcal{J} \nabla \zeta \times \nabla \psi) \cdot \nabla f,$$ (130)

$$\frac{\partial f}{\partial \zeta} = (\mathcal{J} \nabla \psi \times \nabla \theta) \cdot \nabla f.$$ (131)

Using Eq. (128), the directional derivative in the direction of $\nabla \psi$ is written as

$$\nabla \psi \cdot \nabla f = \vert \nabla \psi \vert^2 \frac{\par... ... \theta} + (\nabla \zeta \cdot \nabla \psi) \frac{\partial f}{\partial \zeta} .$$ (132)