Let t be the triangular region with vertices (7, 0, 0), (0, 7, 0), and (0, 0, 7) oriented with an upward-pointing normal vector. assume distances are in meters. a fluid flows with constant velocity field v = 2k meters per second or m/s. calculate the flow rate through t.

calculate the flow rate through the projection of t onto the xy-plane [the triangle with vertices (5, 0, 0), (0, 5, 0), and (0, 0, 0)].

Parameterize by

with both ranging from 0 to 1.

The fluid flows with velocity , so it's flowing upward and we should take the normal vector to to be pointing away from the origin:

Then the flow rate of across is

(in m/s)

The projection of on the plane should have vertices (7, 0, 0), (0, 7, 0), and (0, 0, 0). (unless you're talking about another altogether). Parameterize by

again with both between 0 and 1.

The normal vector is then

and so the flow rate is