Calculus 3

Dive into advanced calculus concepts through a comprehensive 9-hour exploration of multivariable calculus and vector analysis. Begin with an introduction to vectors, covering vector operations, dot and cross products, and their properties. Progress to studying lines and planes in 3D space, including equations, parallelism, and perpendicularity. Explore vector functions, space curves, and their derivatives and integrals. Delve into functions of several variables, limits, partial derivatives, and optimization techniques. Master double and triple integrals, including applications in polar, cylindrical, and spherical coordinates. Conclude with an examination of vector fields, line integrals, Green's Theorem, and an introduction to divergence and curl. Gain a solid foundation in calculus concepts essential for advanced mathematics, physics, and engineering applications.

Introduction to Vectors.
Vector Length and Unit Vectors.
Dot Product.
The Two Definitions of Dot Product Are Equivalent.
Cross Product.
Jusification of the Properties of Cross Product.
Cross Product and Areas of Parallelograms.
More Properties of Cross Product.
Equations of Lines and Planes in 3-D Space.
More Examples of Lines and Planes in 3D.
Parallel and Perpendicular Lines and Planes.
Distances between Points, Lines, and Planes.
Vector Functions and Space Curves.
Closest Point on a Curve.
Derivatives and Integrals of Vector Functions.
Arclength of Parametric Curves.
Functions of Several Variables.
Limits of Functions of Several Variables.
Proof that sin x is smaller in magnitude than x..
Tricky Limit.
Partial Derivatives.
The Chain Rule for Functions of Several Variables.
Justification of the Chain Rule.
Directional Derivatives.
Tangent Plane.
Local Max and Min Values for Functions of Two Variables.
The Second Derivatives Test.
Proof of the Second Derivatives Test.
Lagrange Multipliers.
Double Integrals and Riemann Sums.
Iterated Integrals.
Double Integrals Over General Regions.
Double Integrals in Polar Coordinates.
Justification of the Area Element when Integrating in Polar Coordinates.
The Area under the Normal Curve is 1.
Center of Mass.
Triple Integrals.
Triple Integrals in Cylindrical Coordinates.
Spherical Coordinates.
Triple Integrals in Spherical Coordinates.
Vector Fields.
Line Integrals with respect to Arclength.
Line Integrals in Terms of Riemann Sums.
Line Integrals with Respect to x and y.
Line Integrals and Parametrizations.
Simple Closed Curves - Definitions.
Types of Regions of the Plane - Definitions.
Conservative Vector Fields and Independence of Path.
Green’s Theorem.
Proof of Green’s Theorem.
Parametric Surfaces.
Surface Area of Parametric Surfaces.
Divergence.
Curl.