Gradient

The gradient of a scalar field is $\nabla f=\left(\frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\frac{\partial f}{\partial x_3}\right)$, where the $i$-th component of the gradient is the partial derivative with respect to $x_i$, hence in suffix notation is $[\nabla f{]}_i=\frac{\partial f}{\partial x_i}$. Practically, to find these gradients then you simply calculate each component, often then specifying a value at a specific point.

Example

Find the gradient $\nabla \varphi$ 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 the point $P=(3,1,0)$…

First, the general gradient is $\nabla \varphi =\left(-3x_2,x_2{x}_3^3-3x_1+6x_2^4{x}_3,\frac{3}{2}{x}_2^2{x}_3^2+x_2^5\right)$, by finding the partial derivatives for each component. Then, at the point $P$, we find that $\nabla \varphi {|}_P=(-3,-9,1)=\boxed{-3\mathbf{i}-9\mathbf{j}+\mathbf{k}}$.

Scalar in, vector out