97817_16_ch16_p1158-1164.qk_97817_16_ch16_p1158-1164 11/9/10 9:26 AM Page 1159
SUMMARY
SECTION 16.10
1159
16.10 Summary
The main results of this chapter are all higher-dimensional versions of the Fundamental
Theorem of Calculus. To help you remember them, we collect them together here (without hypotheses) so that you can see more easily their essential similarity. Notice that in
each case we have an integral of a “derivative” over a region on the left side, and the right
side involves the values of the original function only on the boundary of the region.
b
Fundamental Theorem of Calculus
y F⬘共x兲 dx 苷 F共b兲 ⫺ F共a兲
Fundamental Theorem for Line Integrals
y ⵜf ⴢ dr 苷 f 共r共b兲兲 ⫺ f 共r共a兲兲
a
a
b
r(b)
C
C
r(a)
Green’s Theorem
冉
⭸Q
⭸P
C
冊
yy ⭸x ⫺ ⭸y dA 苷 y P dx ⫹ Q dy
D
D
C
n
Stokes’ Theorem
yy curl F ⴢ dS 苷 y F ⴢ dr
S
C
S
C
n
S
Divergence Theorem
yyy div F dV 苷 yy F ⴢ dS
E
S
E
n
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