470 Chapter 16Vectors
We consider here only the special case of a vector field that is the gradient of a scalar
field,v 1 = 1 ∇f. Then
(16.76)
It follows that∇ 1
·
1 ∇ 1 = 1 ∇
2is the Laplacian operator. When a scalar fieldf(r)satisfies
Laplace’s equation ∇
2f 1 = 10 at a point then the derived vector field ∇fhas zero
divergence at the point, and there is no net flux out of a volume element at the point.
This is the case for an incompressible fluid in a region containing no sources (of fluid)
or sinks and for an electrostatic field in a region free of charge.
The curl
The curl(or rotationrot 1 v) of a vector fieldv(r) is defined as the vector
(16.77)
(see Equations (16.52) and (16.53)). In hydrodynamics, the curl of the velocity field at
a point is a measure of the circulation of fluid around the point.
A vector field that is the gradient of a scalar field has zero curl,
curl 1 v 1 = 10 if v 1 = 1 grad 1 f (16.78)
Thus, the electrostatic field is the gradient of the scalar electrostatic potential,
E 1 = 1 −∇φ, so that∇ 1 z 1 E 1 = 10. On the other hand, a vector field that is itself the curl of a
vector field has zero divergence,
div 1 v 1 = 1 0ifv 1 = 1 curl w (16.79)
Thus, the magnetic field can always be expressed as the curl of a vector function,
B 1 = 1
∇
1 z 1 A, where Ais called the magnetic vector potential. Then∇
1
·1 B 1 = 1 0.
0 Exercises 51–55
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