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

Cut here and keep for reference
Chapter 16
13. (a) What is an oriented surface? Give an example of a nonorientable
surface.
An oriented surface S is one for which we can choose a
unit normal vector n at every point so that n varies continuously
over S. The choice of n provides S with an
orientation.
A Möbius strip is a nonorientable surface. (It has only
one side.)
(b) Define the surface integral (or flux) of a vector field F
over an oriented surface S with unit normal vector n.
y
S
y F dS − y
S
y F n dS
(c) How do you evaluate such an integral if S is a parametric
surface given by a vector function rsu, vd?
y
S
y F dS − y
Concept Check Answers (continued)
D
y F sr u 3 r vd dA
We multiply by 21 if the opposite orientation of S is
desired.
(d) What if S is given by an equation z − tsx, yd?
If F − kP, Q, Rl,
y F dS − y y S2P D
Sy −t
−x 2 Q −t
−y 1 R dA
D
for the upward orientation of S; we multiply by 21 for the
downward orientation.
14. State Stokes’ Theorem.
Let S be an oriented piecewise-smooth surface that is bounded
by a simple, closed, piecewise-smooth boundary curve C with
positive orientation. Let F be a vector field whose components
have continuous partial derivatives on an open region in R 3
that contains S. Then
y F dr −
C
yy curl F dS
S
15. State the Divergence Theorem.
Let E be a simple solid region and let S be the boundary surface
of E, given with positive (outward) orientation. Let F be
a vector field whose component functions have continuous
partial derivatives on an open region that contains E. Then
y
S
y F dS − y y
E
y div F dV
16. In what ways are the Fundamental Theorem for Line
Integrals, Green’s Theorem, Stokes’ Theorem, and the
Divergence Theorem similar?
In each theorem, we integrate a “derivative” over a region, and
this integral is equal to an expression involving the values of
the original function only on the boundary of the region.
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