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Gradient, Divergence, and Curl

One approach for implementing differential operators is to use association. Here we associate the standard definitions of the gradient, divergence, and curl (in three-dimensional Cartesian coordinates) to the symbol [Graphics:../Images/index_gr_22.gif]:

[Graphics:../Images/index_gr_23.gif]
[Graphics:../Images/index_gr_24.gif]
[Graphics:../Images/index_gr_25.gif]

We use the symbol [Graphics:../Images/index_gr_26.gif] instead of [Graphics:../Images/index_gr_27.gif] ([Graphics:../Images/index_gr_28.gif]), because [Graphics:../Images/index_gr_29.gif] is a prefix operator, and [Graphics:../Images/index_gr_30.gif] instead of [Graphics:../Images/index_gr_31.gif] ([Graphics:../Images/index_gr_32.gif]).

The set of buttons,

[Graphics:../Images/index_gr_33.gif] [Graphics:../Images/index_gr_34.gif] [Graphics:../Images/index_gr_35.gif]

can be converted into a palette using Generate Palette from Selection under the File menu. Using buttons or palettes simplifies standard vector operator computations. Here we prove that the curl of the gradient of an arbitrary (differentiable) scalar field, [Graphics:../Images/index_gr_36.gif], vanishes.

[Graphics:../Images/index_gr_37.gif]
[Graphics:../Images/index_gr_38.gif]

For an arbitrary (differentiable) vector field:

[Graphics:../Images/index_gr_39.gif]

it is easy to prove vector identities such as

[Graphics:../Images/index_gr_40.gif]
[Graphics:../Images/index_gr_41.gif]

and

[Graphics:../Images/index_gr_42.gif]
[Graphics:../Images/index_gr_43.gif]

This approach is advantageous because

1. it is easy to define rules that closely correspond to usual mathematical notation;
2. the use of a palette or buttons makes application of the rules easy;
3. it is simple to modify these rules to implement operators in different coordinate systems.

An alternative approach is to use the <<Utilities`Notation` package. See the Notation Maker documentation included with the Notebook Demos in the Help Browser.


Converted by Mathematica      September 30, 1999

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