Thomas Calculus 13th [Solutions]

Section 16.8 The Divergence Theorem and a Unified Theory 1217
(b) ( gF) ( gP) ( gN ) i ( gM ) ( gP) j ( gN) ( gM ) k
y z z x x y
g P g N g M g P g N g M
y y z z z z x x x x y y
P g N g i M g P g j N g M g k
g g P N g g M P g g N M
y z y z z x z x x y x y
P N i g g i M P j g g j N M k g g k
g
F
g
F
26. (a) Let F1 M1i N1j P1 k and F2 M 2i N2j P2
k
aF1 bF2 aM1 bM 2 i aN1 bN2 j aP1 bP2
k
( 1 2)
M1 M 2 N1 N2 P1 P2
x x y y z z
aF
bF
a b a b a b
M1 N1 P1 M 2 N2 P2
x y z x y z
a b a F b
(b) Define F1 and F 2 as in part a
a
a
1 2
F
1 2
P1 P2 N1 N2 M1 M 2 P1 P2
y y z z z z x x
aF bF a b a b i a b a b j
N1 N2 M1 M 2
x x y y
a b a b k
P1 N1 M1 P1 N1 M1 P2 N2 M 2 P2 N2 M 2
b
y z z x x y y z z x x y
F
b
1 2
i j k i j k
F
i j k
F F M N P N P P N i M P P M j M N N M k
(c) 1 2 1 1 1 1 2 1 2 1 2 1 2 1 2 1 2
M N P
2 2 2
F1 F2 N1P2 P1 N2 i M1P2 P1 M 2 j M1N2 N1M
2 k
N1P2 P1 N2 M1P x y 2 P1 M 2 M
z 1N2 N1M
2
N1 P2 P1 N2 P2 M1 M 2 P1
x x x x y y y y
P2 N1 N2 P1 M1 P2 P1 M 2
N2 M1 M 2 N1
z z z z
M1 N2 N1 M 2
P1 N1 M1 P1 N1 M1 N2 P2 P2 M 2 M 2 N2
y z z x x y z y x z y x
M 2 N2 P2 M1 N1 P1
F2 F1 F1 F2
27. Let F1 M1i N1j P1
k and F 2 M 2 i N 2 j P 2 k.
(a) F1 F2 N1P2 P1 N2 i P1 M 2 M1P2 j M1N2 N1M
2 k
F F i j
1 2 M1N2 N1M 2 P1 M 2 M1P y z 2 N
z 1P2 P1 N2 M
x 1N2 N1M
2
P1 M 2 M1P x
2 N
y 1P2 P1 N2
k
consider the i-component only: M1N2 N1M 2 P1 M 2 M1P2
y
M1 N2 N1 M 2 P1 M 2 M1 P2
y y y y z z z z
N2 M1 M 2 N1 M 2 P1 P2 M1
M1 M1 M 2 M 2 N2 P2 N1 P1
y z y z y z y z
N2 P2 N1 P1 M1 M 2
z
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