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Figure 1.15 Illustration of gradient.
(1.56)
in a cylindrical coordinate system as
(1.57)
and in a spherical coordinate system as
(1.58)
Example 1.2 Gradient
If function ϕ = x2y + yz is given at point (1,2−1). (a) Find its rate of change for a distance in the direction
Solution
1 Then, at point (1,2,−1)So, its rate of change is found fromwhere
2 The greatest possible rate of change at (1,2,−1) is found fromwhere at (1,2,−1).
1.3.3 Divergence
The divergence of a vector field at a point is a measure of the net outward flux of the same vector per unit volume. The divergence of vector
(1.59)
The divergence can be represented in a cylindrical coordinate system as
(1.60)
In a spherical coordinate system, the divergence is given as
(1.61)
The divergence of a vector field gives a scalar result. In addition, the divergence of a scalar is not a valid operation.
Example 1.3 Divergence
In cylindrical coordinate system, it is given that
Calculate
Solution
