Stokes’ Theorem Divergence Theorem 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 Fundamental theorem of Calculus a b Fundamental Theorem of Line Integrals r(b) r(a) Green’s Theorem C D

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

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

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

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

Stokes’ Theorem: A closer look

Example

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 solid E to a surface integral over the boundary of the solid.

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

Example