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Operators

The foregoing description applies for any function of (x,y,z); the concept of "taking partial derivatives" is independent of what function we are taking the derivatives of. It is therefore practical to learn to think of

$${\partial \over \partial x} \quad \hbox{\rm and} \quad {\pa . . . . . . al y} \quad \hbox{\rm and} \quad {\partial \over \partial y}$$

as OPERATORS that can be applied to any function (like F). Put the operator on the left of a function, perform the operation and you get a partial derivative - a new function of (x,y,z). In general, such "operators" change one function into another. Physics is loaded with operators like these.

The GRADIENT Operator

The GRADIENT operator is a vector operator, written $\Grad{}$ and called "grad" or "del." It is defined (in Cartesian coordinates x,y,z) as¹

$$\Grad{} \; \equiv \; \Hat{\imath} {\partial \over \partial . . . . . . over \partial y} \; + \; \Hat{k} {\partial \over \partial z}$$

and can be applied directly to any scalar function of (x,y,z) - say, $\phi(x,y,z)$ -- to turn it into a vector function, $\Grad{\phi} \; = \; \Hat{\imath} {\partial{\phi} \over \partial x} \; + \; . . . . . . ial{\phi} \over \partial y} \; + \; \Hat{k} {\partial{\phi} \over \partial z}$.