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16.8 and 16.9

16.8 and 16.9. Stokes’ Theorem Divergence Theorem . Important Theorems we know. Fundamental theorem of Calculus. a. b. Important Theorems we know. Fundamental theorem of Calculus. a. b. Fundamental Theorem of Line Integrals. r (b). r (a). Important Theorems we know.

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16.8 and 16.9

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  1. 16.8 and 16.9 Stokes’ Theorem Divergence Theorem

  2. Important Theorems we know • Fundamental theorem of Calculus a b

  3. Important Theorems we know • Fundamental theorem of Calculus a b • Fundamental Theorem of Line Integrals r(b) r(a)

  4. Important Theorems we know • Fundamental theorem of Calculus a b • Fundamental Theorem of Line Integrals r(b) r(a) • Green’s Theorem C D

  5. Important Theorems we know • Fundamental theorem of Calculus a b • Fundamental Theorem of Line Integrals r(b) r(a) Relate an integral of a “derivative” to the original function on the boundary • Green’s Theorem C D

  6. Stokes’ Theorem • A higher dimensional Green’s Theorem • Relates a surface integral over a surfaceS to a line integral around the boundary curve of S

  7. Surface S with boundary C and unit normal vector n n n C (boundary has a positive orientations: Counterclockwise)

  8. Stokes’ Theorem Let S be an oriented piecewise smooth surface that is bounded by a simple, closed piecewise-smooth boundary curve C with positive orientation. Let F be a vector field whose components have continuous first partial derivatives on R3. Then

  9. Stokes’ Theorem: A closer look

  10. Example

  11. The Divergence Theorem • An extension of Green’s Theorem to 3-D solid regions • Relates an integral of a derivative of a function over a solidE to a surface integral over the boundary of the solid.

  12. The Divergence Theorem Let E be a simple solid region and let S be the boundary surface of E, given with positive orientation. Let F be a vector field whose components have continuous first partial derivatives on an open region containing E. Then

  13. Example

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