MATH 251 – LECTURE 32
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: Review
Friday:
webAssign: 14.7, due 4/25 11:55 p.m.
Midterm 3: 14.1–7
Next week: 14.8–9
webAssign: 14.8–9, opens 4/25 12 a.m.
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.
Midterm 3
14.1
14.2
14.3
14.4
14.5
14.6
14.7
Vector fields
Surface integrals of functions and of fields
Conservative vector fields and the Fundamental Theorem
Green’s Theorem
Curl and Divergence
Parametrized surfaces
Surface integrals of functions and of fields
Flux
2
2
RR
Exercise 1. Let S be a the part of the porabolod z = 1 + x + y for which z ≤ 5. Compute the flux
where F is the vector field F = hx, y, zi and n is the normal vector of S with positive z-coordinate.
S
F · dS,
Flux
Exercise 2. The temperature at the point (x, y, z) in a substance with conductivity K = 7.5 is given by
u(x, y, z) = 5y 2 + 5z 2. Find the rate of heat flow inward across the cylindrical surface y 2 + z 2 = 6, 0 ≤ x ≤ 3.
Surfaces
Exercise 3. Let S be the surface with equation x2 + y 2 + z 2 = 4. Find a parametrization of S.
Surfaces
Exercise 4. Let S be the surface with equation x2 + 4y 2 = 4z 2. Find a parametrization of S.
Surfaces
Exercise 5. Let S be the surface with equation z = f (x, y). Find a parametrization of S.
Surface integrals
ZZ
ZZ
F · dS.
f (x, y, z) dS
S
S
Surface integrals
ZZ
f (x, y, z) dS
S
Surface integrals
Exercise 6. Compute the integral
0 ≤ u ≤ 1 and 0 ≤ v ≤ 1.
RR
S
x dS, where S is the surface parametrized by r(u, v) = hu, v, u2i for
Surface integrals
ZZ
F · dS
S
Surface integrals
Exercise 7. Compute the integral
outwards, and F = hx, y, zi.
RR
S
F · dS where S is the cylinder x2 + y 2 = 1 for 0 ≤ z ≤ 1, with n pointing
Surface integrals
Exercise 7. Compute the integral
outwards, and F = hx, y, zi.
RR
S
F · dS where S is the cylinder x2 + y 2 = 1 for 0 ≤ z ≤ 1, with n pointing