MATH 251 – LECTURE 37
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 14.9 and Review
MT. Tuesday is a Friday!
Final exam:
Monday 5/9, at 8–10 am. in BLOC 166
Help Sessions:
Sun–Thu 6–8 p.m. in BLOC 149
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
T 1–2 p.m.
or by appointment.
Divergence
Theorem 1 (The Divergence Theorem). Let E be a simple solid region whose boundary surface S has
positive orientation, and let F be a vector field. Then,
ZZZ
ZZ
div F dV.
F · dS =
S
E
Exercises
Exercise 2. Compute the flux of the vector field F = hx4, −x3z 2, 4xy 2zi across the surface of the solid bounded
by the cylinder x2 + y 2 = 9 and the planes z = x + 7 and z = 0.
Exercises
Exercise 2. Compute the flux of the vector field F = hx4, −x3z 2, 4xy 2zi across the surface of the solid bounded
by the cylinder x2 + y 2 = 9 and the planes z = x + 7 and z = 0.
Exercises
Exercise 3. Compute the surface integral
half of the sphere x2 + y 2 + z 2 = 9.
3
y
2
2
2
S F · dS, where F = hz x, 3 + sin(z), x z + y i and S is the top
RR
Exercises
Exercise 3. Compute the surface integral
half of the sphere x2 + y 2 + z 2 = 9.
3
y
2
2
2
S F · dS, where F = hz x, 3 + sin(z), x z + y i and S is the top
RR
Exercises
Exercise 4. Evaluate
and 0 ≤ z ≤ 1.
RR
S
F · dS, where F = hx2, y 3, z 4i and S is the surface of the cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
Exercises
Exercise 4. Evaluate
and 0 ≤ z ≤ 1.
RR
S
F · dS, where F = hx2, y 3, z 4i and S is the surface of the cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1
Exercises
Exercise 5. Let E be the solid cylinder with equations 0 RR
≤ x2 + y 2 ≤ R2 and 0 ≤ z ≤ 1. Let S be the surface
of E, and let F be the vector field hx3, y 3, z 3i. Evaluate
F · dS.
Exercises
Exercise 5. Let E be the solid cylinder with equations 0 RR
≤ x2 + y 2 ≤ R2 and 0 ≤ z ≤ 1. Let S be the surface
of E, and let F be the vector field hx3, y 3, z 3i. Evaluate
F · dS.