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##

Gradient field.

A *vector field* in the plane is a function which assigns to every point in a domain
in the plane a vector
(v.s subsection 6.9.

A *vector field* in the 3-dimensional space is a function which assigns to every point in a domain
in the space a vector
.

The field is *continuous* (resp. *differentiable*) if the components *M*,*N*,*P* are continuous (resp. differentiable).

**Definition 8.8**
Let

*f* be a differentiable function of three variables over a domain

in the space.
The

*gradient field* of

*f* is the field of gradient vectors

**Example 8.9**
Let

*f*(

*x*,

*y*,

*z*)=

*xy*+

*z*^{2}. Then:

The gradient field of

*f* is given by:

**Example 8.10**
Let

.
Then:

The gradient field of

*f* is given by:

*Noah Dana-Picard*

*2001-05-30*