c Dr Oksana Shatalov, Spring 2013
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Spring 2013 Math 251
Week in Review 12
courtesy: Oksana Shatalov
(covering Sections 14.8 (continued), 14.9 )
14.8: Stokes’ Theorem (continued)
Key Points
• Let S be an oriented piece-wise-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 R3 that contains S. Then
I
ZZ
F · dr =
curlF · dS.
C
S
Examples
I
1. Evaluate I =
F · dr if F = h2y + 3ex , z − y 8 , x + ln(z 2 + 1)i and C is the curve of inter-
C
section of the plane x+y+z = 0 and sphere x2 +y 2 +z 2 = 1. (Orient C to be counterclockwise
when viewed from above).
2. Verify Stokes’ Theorem for the surface S: x2 + y 2 + 5z = 1, z ≥ −5 (oriented by upward
normal) and the vector field F~ = xz~i + yz~j + (x2 + y 2 )~k.
RR
R
~=
3. Verify Stokes’ Theorem S curlF~ · dS
F~ · d~r for the vector field F~ = hyz 2 , −xz 2 , z 3 i
∂S
and the cylinder x2 + y 2 = 9 for 1 ≤ z ≤ 2 oriented out.
14.9: The Divergence Theorem
Key Points
• The Divergence Theorem: Let E be a simple solid region whose boundary surface S has positive (outward) orientation. Let F be a continuous vector field on an open region that contains E. Then
ZZ
ZZZ
F · dS =
divF dV.
S
E
RR
~ where F~ = x3~i + y 3~j + z 3~k and S is
4. Use the Divergence Theorem to compute S F~ · dS,
the surface of the region enclosed by x2 + y 2 = 1 and the planes z = 0, z = 2.
5. (cf. problem 4, WIR #11) Use the Divergence Theorem to find flux of the vector field
F = hx, y, 1i across the surface S which is the boundary of the region enclosed by the
cylinder y 2 + z 2 = 1 and the planes x = 0 and x + y = 5.
c Dr Oksana Shatalov, Spring 2013
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6. Verify the Divergence Theorem for the region
E = {(x, y, z) : 0 ≤ z ≤ 9 − x2 − y 2 } and the vector field F~ = ~r = x~i + y~j + z~k
RR
7. Apply the Divergence Theorem to compute S F · dS for the vector field
F(x, y, z) = hx3 + sin(yz), y 3 , y + z 3 i
over the complete boundary S of the solid hemisphere {(x, y, z) : x2 + y 2 + z 2 ≤ 1, z ≥ 0}
with outward normal.