Lecture 6: Surface Integrals. Recall, area is a vector. Vector area of this surface is which has sensible magnitude and direction Or, by projection . z. (0,1,1). (1,0,1). y. (0,1,0). x. (1,0,0). z. z. x. y. y. (1,0). x. Surface Integrals.
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which has sensible magnitude and direction
Do x first
Then do y
Example: volume flow of fluid, with velocity field , through a surface element yielding a scale quantity
Consider an incompressible fluid
“What goes into surface S through S1 comes out through surface S2”
S1 and S2 end on same rim
and depends only on rim not surface
(see second sheet of this Lecture)
want to calculate
Note S need not be planar!
Note also, project onto easiest plane
and the disk S2
Exercises for student (in polars over the unit circle, i.e. projection of both S1 and S2 onto xz plane)