Dive into a comprehensive exploration of vector calculus through a series of video lectures designed for advanced calculus students. Learn about parametric curves, line integrals, vector fields, and fundamental theorems in multivariable calculus. Explore concepts such as the gradient, divergence, and curl of vector fields, and master techniques for solving line and surface integrals. Discover important theorems like Green's Theorem, the Divergence Theorem, and Stokes' Theorem, along with their applications. Gain insights into parametric surfaces, surface area calculations, and flux computations. Enhance your understanding of vector calculus through numerous examples and practical applications, preparing you for advanced mathematical analysis in physics and engineering.
Overview
Syllabus
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.
Taught by
Sean Fitzpatrick
