Grad Operator

It is useful to define the vector operator

$$\nabla \equiv \left( \frac{\partial}{\partial x},\, \frac{\partial}{\partial y},\, \frac{\partial }{\partial z}\right),$$ (1362)

which is usually called the grad or del operator. This operator acts on everything to its right in a expression, until the end of the expression or a closing bracket is reached. For instance,

$${\bf grad}\,f = \nabla f \equiv \left(\frac{\partial f}{\par... ...partial f}{\partial y},\,\frac{\partial f}{\partial z}\right).$$ (1363)

For two scalar fields $\phi$ and $\psi$,

$${\bf grad}\,(\phi \,\psi) = \phi\,\, {\bf grad}\,\psi +\psi\,\, {\bf grad}\,\phi$$ (1364)

can be written more succinctly as

$$\nabla(\phi\, \psi) = \phi \,\nabla\psi + \psi\, \nabla \phi.$$ (1365)

Suppose that we rotate the coordinate axes through an angle $\theta$ about $Oz$. By analogy with Equations (A.1277)-(A.1279), the old coordinates ($x$, $y$, $z$) are related to the new ones ($x'$, $y'$, $z'$) via

$$x = x'\,\cos\theta - y'\,\sin \theta,$$ (1366)

$$y = x\,'\sin\theta +y'\,\cos\theta,$$ (1367)

$$z = z'.$$ (1368)

Now,

$$\frac{\partial}{\partial x'} = \left(\frac{\partial x}{\part... ...l z}{\partial x'} \right)_{y',z'} \frac{\partial}{\partial z},$$ (1369)

giving

$$\frac{\partial}{\partial x'} = \cos\theta \,\frac{\partial}{\partial x} + \sin\theta \,\frac{\partial}{\partial y},$$ (1370)

and

$$\nabla_{x'} = \cos\theta\, \nabla_x + \sin\theta \,\nabla_y.$$ (1371)

It can be seen, from Equations (A.1280)-(A.1282), that the differential operator $\nabla$ transforms in an analogous manner to a vector. This is another proof that $\nabla f$ is a good vector.

Figure A.113: Cylindrical polar coordinates.