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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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lecture 6 surface integrals
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

surface integrals

y

(1,0)

x

Surface Integrals
  • Example from Lecture 3 of a scalar field integrated over a small (differential) vector surface element in plane z=0
  • Example

(0,1)

Do x first

Then do y

  • But answer is a vector ; get positive sign by applying right-hand rule to an assumed path around the edge (z>0 is “outside” of surface)
vector surface integrals
Vector Surface Integrals

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

examples
Examples
  • Consider a simple fluid velocity field
  • Through the triangular surface
  • More complicated fluid flow requires an integral

(0,1)

y

(1,0)

x

(see second sheet of this Lecture)

evaluation of surface integrals by projection
Evaluation of Surface Integrals by Projection

want to calculate

z

y

x

because

Note S need not be planar!

Note also, project onto easiest plane

example
Example
  • Surface Area of some general shape
another example
Another Example
  • and S is the closed surface given by the paraboloid S1

and the disk S2

  • On S1 :

Exercises for student (in polars over the unit circle, i.e. projection of both S1 and S2 onto xz plane)

  • On S2 :
  • So total “flux through closed surface” (S1+S2) is zero
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