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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)

Description:

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. Read more

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

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#Mathematics #Calculus #Vector Calculus #Green's Theorem #Stokes' Theorem #Vector Fields #Surface Integrals #Divergence Theorem #Line Integrals