Del

The del operator \(\nabla\) is an easy way to write certain values in multivariable calculus. We can write this operator in symbolic notation (not a real vector) to understand how it works:

\[ \nabla = \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \rangle \]

Thus, the gradient of a function \(f\) can be written as \(\nabla f\):

\[ \nabla f = \langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \rangle \]

Similarly, the divergence of a function \(f=\langle P,Q,R \rangle\) can be written as \(\nabla \cdot f\):

\[ \nabla \cdot f = \langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \rangle \cdot \langle P,Q,R \rangle = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} + \frac{\partial R}{\partial z} \]