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

Section 16.10 Summary 1147
The main results of this chapter are all higher-dimensional versions of the Funda mental
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.
Fundamental Theorem of Calculus
y b
F9sxd dx − Fsbd 2 Fsad
a
a
b
Fundamental Theorem for Line Integrals
y C
=f dr − f srsbdd 2 f srsadd
a
r (a)
a
a
r (a)
a
r (a)
r (a)
C
C
D C
C
C
C
r (b)
b
r
b
(b)
b
r (b)
b
r (b)
r (b)
r (a)
D C
C
Green’s Theorem
yy S −Q
−x 2 −P dA −
−yD y P dx 1 Q dy
C
D
C
D n
C
D
C
S
D
n
S n
Stokes’ Theorem
yy curl F dS − y F dr
C
S
n
C n
S
n
S
C
E
n
C S
C
E n
n
S
n
S
n
S
Divergence Theorem
y yy div F dV − y
E
S
y F dS
E n
E
E
S
n
S
n
n
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