MATH 251 – LECTURE 36
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 14.8–9
webAssign: 14.8–9, due 5/2 11:55 p.m.
Next week: 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.
F 1–2 p.m.
or by appointment.
Divergence
2
Exercise 1. Let F = hyez , y 2, exy i. Compute the divergence of F .
Divergence
Z
ZZ
Q0x
F · dr =
C
D
−
Py0
dx dy
Divergence
Theorem 2 (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
Divergence
RR
2
Exercise 3. Compute the surface integral S F · dS, where F = hyez , y 2, exy i and S is the surface of the
solid bounded by the cylinder x2 + y 2 = 9 and the planes z = 0 and z = y − 3.
Divergence
RR
2
Exercise 3. Compute the surface integral S F · dS, where F = hyez , y 2, exy i and S is the surface of the
solid bounded by the cylinder x2 + y 2 = 9 and the planes z = 0 and z = y − 3.
Divergence
RR
Exercise 4. Compute the surface integral S F · dS, where F = hx2z 3, xyz 3, xz 4i and S is the surface of the
box with vertices (±3, ±3, ±2) with positive orientation.
Divergence
RR
Exercise 5. Compute the surface integral S F · dS, where F = hx2z 3, xyz 3, xz 4i and S is the surface of the
box with vertices (±3, ±3, ±2) except the bottom, with positive orientation.
Divergence
Theorem 3 (The Divirgence 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
Corollary
6. Let S be a simple closed surface, and let F be a vector field such that div F = 0. Then
RR
S F · dS = 0.
Divergence
RR
Exercise 7. Compute the surface integral S F · dS, where F = hx2y, xy 2, 4xyzi and S is the surface of the
simplex bounded by the planes x = 0, y = 0, z = 0, and x + 2y + z = 2.
Divergence
RR
Exercise 7. Compute the surface integral S F · dS, where F = hx2y, xy 2, 4xyzi and S is the surface of the
tetrahedron bounded by the planes x = 0, y = 0, z = 0, and x + 2y + z = 2.