MATH-M 312 Lecture Notes - Lecture 12: Unit Disk, Parallelogram, Surface Integral
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M312 sections 8. 1, 8. 2, and 8. 4 notes- the fundamental theorems of vector calculus (the divergence theorem, g(cid:396)ee(cid:374)"s. Recall that (cid:1516) (cid:1858)(cid:4666)(cid:1876)(cid:4667) (cid:1856)(cid:1876)=(cid:1832)(cid:4666)(cid:1854)(cid:4667) (cid:1832)(cid:4666)(cid:1853)(cid:4667) and (cid:1516) (cid:3031)(cid:3033)(cid:3031)(cid:3051)(cid:3029)(cid:3028) (cid:3029)(cid:3028) If (cid:2198) and (cid:2199) are endpoints of oriented curve (cid:1855), call (cid:2198) start and (cid:2199) end; then (cid:1516) (cid:1858) (cid:1856)(cid:1871) (cid:3030) Tokes" theore(cid:373)- for any vector field (cid:2162): (cid:2871) (cid:2871) and oriented surface in (cid:2871): (cid:1517) (cid:1855)(cid:1873)(cid:1870)(cid:1864)(cid:4666)(cid:2162)(cid:4667) (cid:1856) If a person is standing on boundary in direction given by orientation of (outward/up), orientation of (cid:2034) is the one that keeps o(cid:374) pe(cid:396)so(cid:374)"s left as he (cid:449)alks. Say we have oriented surface with normal vectors on back side. Divide into small parallelograms (all oriented the same way) Calculate (cid:1516) (cid:2162) (cid:1856) for each parallelogram, add up all line integrals over. Each line segment is part of line integrals over 2 parallelograms; since the same segment has opposite orientations: proof, ex.
