James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 14.3 Partial Derivatives 921
Partial differential equations involving functions of three variables are also very
important in science and engineering. The three-dimensional Laplace equation is
5
− 2 u
−x 2
1 −2 u
−y 2
1 −2 u
−z 2 − 0
and one place it occurs is in geophysics. If usx, y, zd represents magnetic field strength at
position sx, y, zd, then it satisfies Equation 5. The strength of the magnetic field indicates
the distribution of iron-rich minerals and reflects different rock types and the location of
faults. Figure 9 shows a contour map of the earth’s magnetic field as recorded from an
aircraft carrying a magnetometer and flying 200 m above the surface of the ground. The
contour map is enhanced by color-coding of the regions between the level curves.
FIGURE 9
Magnetic field strength of the earth
Saskatchewan Ministry of Energy and Resources / Roger Watson
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Figure 10 shows a contour map for the second-order partial derivative of u in the vertical
direction, that is, u zz . It turns out that the values of the partial derivatives u xx and u yy
are relatively easily measured from a map of the magnetic field. Then values of u zz can
be calculated from Laplace’s equation (5).
FIGURE 10
Second vertical derivative
of the magnetic field
Saskatchewan Ministry of Energy and Resources / Roger Watson
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