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Next: GRADIENTS of Scalar Functions Up: Vector Calculus Previous: Functions of Several Variables

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

\begin{displaymath}{\partial \over \partial x} \quad \hbox{\rm and} \quad
{\pa . . . 
 . . . al y} \quad \hbox{\rm and} \quad
{\partial \over \partial y} \end{displaymath}

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

\begin{displaymath}\Grad{} \; \equiv \;
\Hat{\imath} {\partial \over \partial  . . . 
 . . . over \partial y}
\; + \; \Hat{k} {\partial \over \partial z}
\end{displaymath}

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, $\Grad{\phi} \; = \;
\Hat{\imath} {\partial{\phi} \over \partial x}
\; + \;  . . . 
 . . . ial{\phi} \over \partial y}
\; + \; \Hat{k} {\partial{\phi} \over \partial z}$.



Jess H. Brewer - Last modified: Wed Nov 18 12:22:08 PST 2015