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Grad Operator
It is useful to define the vector operator

(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,

(1363) 
For two scalar fields and ,

(1364) 
can be written more succinctly as

(1365) 
Suppose that we rotate the coordinate axes through an angle about .
By analogy with Equations (A.1277)(A.1279), the old coordinates (, , ) are related
to the new ones (, , ) via
Now,

(1369) 
giving

(1370) 
and

(1371) 
It can be seen, from Equations (A.1280)(A.1282), that
the differential operator transforms in an analogous manner to
a vector.
This is another proof that is a good vector.
Figure A.113:
Cylindrical polar coordinates.

Next: Curvilinear Coordinates
Up: Vector Algebra and Vector
Previous: Gradient
Richard Fitzpatrick
20110331