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Stokes’ Theorem Divergence Theorem
16.8 and 16.9 Stokes’ Theorem Divergence Theorem
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Important Theorems we know
Fundamental theorem of Calculus a b
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Important Theorems we know
Fundamental theorem of Calculus a b Fundamental Theorem of Line Integrals r(b) r(a)
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Important Theorems we know
Fundamental theorem of Calculus a b Fundamental Theorem of Line Integrals r(b) r(a) Green’s Theorem C D
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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
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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
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C (boundary has Surface S with boundary C and unit normal vector n n n
a positive orientations: Counterclockwise)
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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
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Stokes’ Theorem: A closer look
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Example
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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.
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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
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Example
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