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

920 Chapter 14 Partial Derivatives
and using Clairaut’s Theorem it can be shown that f x yy − f yx y − f yyx if these functions are
continuous.
EXAMPLE 8 Calculate f xx yz if f sx, y, zd − sins3x 1 yzd.
SOLUTION
f x − 3 coss3x 1 yzd
f xx − 29 sins3x 1 yzd
f xx y − 29z coss3x 1 yzd
f xx yz − 29 coss3x 1 yzd 1 9yz sins3x 1 yzd
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Partial Differential Equations
Partial derivatives occur in partial differential equations that express certain physical
laws. For instance, the partial differential equation
− 2 u
−x 2
1 −2 u
−y 2 − 0
is called Laplace’s equation after Pierre Laplace (1749–1827). Solutions of this equa -
tion are called harmonic functions; they play a role in problems of heat conduction,
fluid flow, and electric potential.
EXAMPLE 9 Show that the function usx, yd − e x sin y is a solution of Laplace’s equation.
SOLUTION We first compute the needed second-order partial derivatives:
u x − e x sin y u y − e x cos y
u xx − e x sin y u yy − 2e x sin y
So u xx 1 u yy − e x sin y 2 e x sin y − 0
Therefore u satisfies Laplace’s equation.
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The wave equation
− 2 u
−t 2
− a2 −2 u
−x 2
x
FIGURE 88
u(x, t)
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describes the motion of a waveform, which could be an ocean wave, a sound wave, a
light wave, or a wave traveling along a vibrating string. For instance, if usx, td represents
the displacement of a vibrating violin string at time t and at a distance x from one end of
the string (as in Figure 8), then usx, td satisfies the wave equation. Here the constant a
depends on the density of the string and on the tension in the string.
EXAMPLE 10 Verify that the function usx, td − sinsx 2 atd satisfies the wave equation.
SOLUTION u x − cossx 2 atd u t − 2a cossx 2 atd
u xx − 2sinsx 2 atd
u tt − 2a 2 sinsx 2 atd − a 2 u xx
So u satisfies the wave equation.
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