Calculus IV - Vector Calculus - Line Integrals, Surface Integrals, Vector Fields, Green's Theorem, Divergence Theorem, Stokes Theorem

Dive into a comprehensive 5-hour course on vector calculus, covering approximately six weeks of advanced calculus topics. Explore line integrals, surface integrals, vector fields, conservative fields, Green's theorem, divergence theorem, and Stokes' Theorem. Begin with an introduction to vector calculus and progress through curve parameterizations, arclength parameterization, and various types of integrals. Learn to sketch and analyze vector fields, understand gradient vector fields, and compute line integrals of vector fields. Investigate flow integrals, circulation, and flux integrals. Delve into conservative vector fields, the fundamental theorem of line integrals, and methods for testing and finding scalar potential functions. Study curl, divergence, and their relationships to Green's Theorem. Examine surface descriptions, area formulas, and integrals for parametric, implicit, and explicit surfaces. Discover the concepts of orientable and non-orientable surfaces, including the Mobius strip. Master the application of Stokes' Theorem and the Divergence Theorem, including real-world applications like Gauss's Law for Electric Flux. Conclude with a unified view of vector calculus, connecting all major theorems and concepts covered throughout the course.

What is VECTOR CALCULUS?? **Full Course Introduction**.
Curves, Parameterizations, and the Arclength Parameterization.
What is a LINE INTEGRAL? // Big Idea, Derivation & Formula.
Line Integrals: Full Example.
Line Integrals in 3D // Formula & Three Applications.
Intro to VECTOR FIELDS // Sketching by hand & with computers.
The Gradient Vector Field.
Line Integrals of Vector Fields // Big Idea, Definition & Formula.
Example: Computing the Line Integral of a Vector Field (i.e. Work Done).
Line Integrals with respect to x or y // Vector Calculus.
Flow Integrals and Circulation // Big Idea, Formula & Examples // Vector Calculus.
Flux Integrals // Big Idea, Formula & Examples // Vector Calculus.
Conservative Vector Fields // Vector Calculus.
The Fundamental Theorem of Line Integrals // Big Idea & Proof // Vector Calculus.
How to Test if a Vector Field is Conservative // Vector Calculus.
Finding the scalar potential function for a conservative vector field // Vector Calculus.
Curl or Circulation Density of a Vector Field // Vector Calculus.
Curl, Circulation, and Green's Theorem // Vector Calculus.
Divergence, Flux, and Green's Theorem // Vector Calculus.
Example: Using Green's Theorem to Compute Circulation & Flux // Vector Calculus.
Describing Surfaces Explicitly, Implicitly & Parametrically // Vector Calculus.
The Surface Area formula for Parametric Surfaces // Vector Calculus.
Why is the surface area of a Sphere 4pi times radius squared???.
Computing the Surface Area of a surface parametrically // Example 1 // Vector Calculus.
Computing the Surface Area of a surface parametrically // Example 2 // Vector Calculus.
Surface Area for Implicit & Explicit Surfaces // Vector Calculus.
Computing the Surface Area of an Implicitly Defined Surface.
Surface Integrals // Formulas & Applications // Vector Calculus.
Orientable vs Non-Orientable Surfaces and the Mobius Strip.
Flux of a Vector Field Across a Surface // Vector Calculus.
Computing the Flux Across a Surface // Vector Calculus.
The CURL of a 3D vector field // Vector Calculus.
Stokes' Theorem // Geometric Intuition & Statement // Vector Calculus.
Stokes' Theorem Example // Verifying both Sides // Vector Calculus.
The Divergence Theorem // Geometric Intuition & Statement // Vector Calculus.
Divergence Theorem example: Flux across unit cube // Vector Calculus.
Divergence Theorem for regions bounded by two surfaces // Vector Calculus.
Deriving Gauss's Law for Electric Flux via the Divergence Theorem from Vector Calculus.
A unified view of Vector Calculus (Stoke's Theorem, Divergence Theorem & Green's Theorem).