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Parametric curves and arc length.
Line integrals of functions.
Example: line integral of a function.
Introducing vector fields.
The 'del' operator.
The gradient vector field.
Divergence of a vector field.
Curl of a vector field.
Some divergence examples.
Interpreting curl.
Line integrals of scalar fields, revisited.
Line integrals of vector fields.
Line integral example 1.
Line integral example 2.
Terminology for curves.
Properties of line integrals.
Fundamental Theorem of Calculus.
FTC example 1.
FTC example 2.
FTC example 3.
FTC example 4.
Conservative vector fields and independence of path.
Further terminology for curves in the plane.
Green's Theorem.
Verifying Green's Theorem with an example.
Sketching the proof of Green's Theorem.
Green's Theorem for multiply-connected domains: an example.
Using Green's Theorem to find the area under a cycloid.
Green's Theorem and polygon areas.
Parametric surfaces.
Tangent and normal vectors for parametric surfaces; surface area.
Surface area.
Surface area of a sphere.
Surface integral of a scalar field.
Surface integrals of vector fields.
Surface integral example.
Computing flux across a cube (surface with 6 faces!).
Some surface integrals over spheres.
The Divergence Theorem.
Gauss' Law.
Stokes' Theorem.
Using Stokes' Theorem: Example 1.
Using Stokes' Theorem: Example 2.
Sean Fitzpatrick
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