Question

Verify equation (6.8); that is, find∇fin spherical coordinates as we did for cylindrical coordinates.

Short answer

$\nabla f=e_r\frac{\partial f}{\partial r}+e_{\theta }\frac{1}{r}\frac{\partial f}{\partial \theta }+e_z\frac{\partial f}{\partial z}$

Step-by-step solution

Given Information

The given information is the spherical coordinates to be used in solving the question.

Definition of Scalar field.

In mathematics and physics, scalar field or scalar-valued function referred to a scalarvalueto everypointin aspace– possiblyphysical space. The scalar may either be a (dimensionless)mathematical numberor aphysical quantity.

Find the solution.

Let $f\left(r,\theta ,\varphi \right)$ be the gradient of a general function in a spherical coordinates.

The gradient is the directional derivative in the form of $\frac{df}{ds}$ .

The arc length is equal to which is in the direction of $e_r$ , which further leads to the derivative $\frac{df}{dr}$.

In the direction of $e_{\theta }$ the arc length is$rd\theta$ , which further leads to the derivative$\frac{df}{r\partial \theta }$ .

Since the angle $\varphi$ is obtained by rotating around k-axis. The arc length is therefore $r\sin \theta d\varphi$, which gives the directional derivatives in the direction of $e_{\theta }$as $\frac{\partial f}{r\sin \theta \partial \varphi }$ .

Therefore, the gradient is $\nabla f=e_r\frac{\partial f}{\partial r}+e_{\theta }\frac{1}{r}\frac{\partial f}{\partial \theta }+e_z\frac{\partial f}{\partial z}$ .

Hence, the solution to this problem is $\nabla f=e_r\frac{\partial f}{\partial r}+e_{\theta }\frac{1}{r}\frac{\partial f}{\partial \theta }+e_z\frac{\partial f}{\partial z}$.