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Figure 3.10: Gradient of a Vector

The quantity .Dv(x)-m is called the directional derivative because it gives the rate of change of the vector field as we move in the direction m.

■ Example 3-5: Gradient of a Vector Field

Determine the gradient of the following vector field v(x) = xix2x3(xiei + x2e2 + x3e3) Solution:

2xix2x3 [ei ® ei] + xix3 [ei ® e2] + x2x2 [ei ® e3] +x2x3 [e2 ® ei] + 2xix2x3[e2 ® e2] + xix2 [e2 ® e3] +x2x3[e3 ® ei] + xix3[e3 ® e2] + 2xix2x3 [e3 ® e3] 2 xi/x2 xi/x3 = xix2x3 x2/xi 2 x2/x3 x3/xi x3/x2 2

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