Velocity Field

The velocity field $\mathbf{V}=\mathbf{V}(x,y,z,t)$ of a fluid is given as

$$\begin{array}{rl}\mathbf{V}& =u\mathbf{i}+v\mathbf{j}+w\mathbf{k}\\ & =u(x,y,z,t)\mathbf{i}+v(x,y,z,t)\mathbf{j}+w(x,y,z,t)\mathbf{k}\end{array}$$

The magnitude can be found with

$$V=|\mathbf{V}|=\sqrt{u^2+v^2+w^2}$$