Gradient

Relevant parts to questions...

• In suffix form: $[\nabla f{]}_i=\partial f/\partial x_i$.
• Radial function: $\nabla f(r)=f^{\prime }(r)\mathbf{r}/r$, derived from $\partial r/\partial x_i=x_i/r$.
• Tensor character: $\nabla f$ is a rank-1 tensor because $\partial /\partial x_i^{\prime }=L_{ij}\partial /\partial x_j$ (chain rule + Cartesian Jacobian).

The gradient of a scalar field is

$$\nabla f=\left(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\frac{\partial f}{\partial x_3}\right)$$

In suffix notation: $[\nabla f{]}_i=\partial f/\partial x_i$. To compute, just take partial derivatives component-by-component, optionally evaluated at a point.

Example

Find $\nabla \varphi$ of $\varphi (x_1,x_2,x_3)=\frac{1}{2}{x}_2^2{x}_3^3-3x_1{x}_2+x_2^5{x}_3+1$ at $P=(3,1,0)$.

$\nabla \varphi =\left(-3x_2,x_2{x}_3^3-3x_1+5x_2^4{x}_3,\frac{3}{2}{x}_2^2{x}_3^2+x_2^5\right)$ At $P$: $\boxed{\nabla \varphi {|}_P=-3\mathbf{i}-9\mathbf{j}+\mathbf{k}}$.

Radial Functions $f(r)$

For $r=\Vert \mathbf{r}\Vert =(x_j{x}_j{)}^{1/2}$:

$$\frac{\partial r}{\partial x_i}=\frac{1}{2}(x_j{x}_j{)}^{-1/2}\cdot 2x_i=\frac{x_i}{r}$$

So by chain rule, $[\nabla f(r){]}_i=f^{\prime }(r)\frac{\partial r}{\partial x_i}=f^{\prime }(r)\frac{x_i}{r}$, i.e.,

$$\boxed{\nabla f(r)=f^{\prime }(r)\frac{\mathbf{r}}{r}}$$

Scalar in, vector out.