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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. y. (1,0). x. Surface Integrals.

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Lecture 6: Surface Integrals

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  1. 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

  2. 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)

  3. 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

  4. 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)

  5. 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

  6. Example • Surface Area of some general shape

  7. 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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