MATH 251 – LECTURE 31
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 14.5–7
webAssign: 14.5–6, due 4/18 11:55 p.m.
Next week: 14.7
Friday:
webAssign: 14.7, opens 4/18 12 a.m.
Midterm 3: 14.1–7
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.
Flux
Exercise 1. Let S be a surface parametrized by r(u, v) = heu, ev , u + vi. Comput a normal vector to S at
r(u, v). Compute the normalized normal vector to S at r(u, v).
Flux
Let S be a surface. Let n denote a unit normal to S.
Definition 2. The surface S is said to be orientable if n can be chosen so that it varies continuously along S.
Flux
Definition 3. Let S be a orientable surface with unit normal n. Let F we a vector field. Then, the flux (or
total flow) of F in the direction of n across S is
ZZ
ZZ
F · n dS
F · dS =
S
S
Flux
Exercise 4. Let F = hx2y, −3xy 2, 4y 3i. Let S be the part of the elliptic paraboloid z = x2 + y 2 − 9 that lies
below the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, oriented so that the normal vector points downwards. Compute
RR
S F · dS.
Flux
Exercise 4. Let F = hx2y, −3xy 2, 4y 3i. Let S be the part of the elliptic paraboloid z = x2 + y 2 − 9 that lies
below the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 1, oriented so that the normal vector points downwards. Compute
RR
S F · dS.
Flux
A closed orientable surface is said to be positively oriented if the normal vector n always points outwards.
MOVE THIS DEFINITION TO THE NEXT LECTURE
Flux
Exercise 4. A fluid has velocity field v = h−y, x, 2zi. Compute the rate of flow through the sphere x2 +y 2 +z 2 =
25 with positive orientation.
BAD EXAMPLE
Flux
Exercise 4. A fluid has velocity field v = h−y, x, 2zi. Compute the rate of flow through the sphere x2 +y 2 +z 2 =
25 with positive orientation.
BAD EXAMPLE