MATH 251 – LECTURE 34
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.
W 2–3 p.m.
or by appointment.
Curve integrals in three dimensions
Exercise 1. Comput the integral
and F = h1, z, xi.
R
C
F · dr , where C is the parametrized by r(t) = ht, t2, t/2i, when 0 ≤ t ≤ 1,
Green’s theorem revisited
Z
ZZ
Q0x
F · dr =
C
D
−
Py0
dxdy
Stoke’s theorem
Definition 2. Let S be a orientable surface with simple, closed, smooth boundary, and let n be a normal
vector of S. Then, the boundary has positive orientation if, walking along the boundary on the same side of the
normal, the surface is on your left hand side.
Stoke’s theorem
Theorem 3. 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
RR
Exercise 4. Use Stoke’s theorem to evaluate S curl(F )·dS if F = hxyz, x, exy cos(z)i where S is the northern
hemisphere of x2 + y 2 + z 2 = 1, with n as the upwards pointing normal.
Stoke’s theorem
R
Exercise 5. Use Stoke’s theorem to evaluate C F · dr if F = hz 2, y 2, xyi, and C is the triangle with vertices
(1, 0, 0), (0, 1, 0), and (0, 0, 2) oriented counterclockwise as viewed from above.
Stoke’s theorem
R
Exercise 5. Use Stoke’s theorem to evaluate C F · dr if F = hz 2, y 2, xyi, and C is the triangle with vertices
(1, 0, 0), (0, 1, 0), and (0, 0, 2) oriented counterclockwise as viewed from above.
Stoke’s theorem
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.