MATH 251 – LECTURE 35
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.
Stoke’s Theorem
Theorem 1. Let S be a piecewise smooth orientable surface with a simple, closed boundary curve C. Let
n be a normal vector of S such that C is positively oriented. Let F be a vector field. Then
ZZ
Z
curl(F ) · dS
F · dr =
C
S
Stoke’s Theorem
Corollary 2. Let F be a vector field such that curl(F ) = h0, 0, 0i. Let C be a closed curve. Then
Z
F · dr = 0.
C
Exercise 3. Let F = h2xy, x2, zi. Compute the integral
R
C
F · dr where C is the unit circle in the yz-plane.
Stoke’s theorem
Corollary 4. Let S be a piecewise smooth orientable closed surface, and let F be a vector field. Then,
ZZ
curl(F ) · dS = 0.
S
Exercise 5. Let F be the vector field
F = he
Compute the integral
RR
S
cos(x+z sin(y))
27x
, sin
√
(arctan(y +
4 + z 4)), π 6xyz i.
curl(F ) · dS where S is the unit sphere with positive orientation.
Stoke’s Theorem
RR
Exercise 6. Compute the flux integral S curl(F ) · dS where F = hx, x2, x3i and S is the surface of the unit
cube except bottom with normal vector pointing outwards.
Stoke’s Theorem
Corollary 7. Let S1 and S2 be two piecewise smooth orientable closed surfaces whith the same oriented
boundary C. Let n1 and n2 be unit normals of S1 resp. S2 such that C is positively oriented with respect
to each of them. Then
ZZ
ZZ
curl(F ) · dS =
S1
curl(F ) · dS.
S2