Calculus III - Multivariable Calculus - Vectors, Curves, Partial Derivatives, Multiple Integrals, Optimization

Overview

This course on Multivariable Calculus aims to teach students the foundational knowledge required for understanding vectors, curves, partial derivatives, multiple integrals, and optimization. The course emphasizes geometric and conceptual understanding of the topics covered. Students will learn about vectors, lines, planes, vector-valued functions, partial derivatives, integration, curvature, arclength, optimization, and more. The teaching method includes video lectures that explain the concepts with a focus on geometric interpretations. This course is intended for learners interested in advancing their knowledge of calculus beyond the single-variable level.

Syllabus

What are the big ideas of Multivariable Calculus?? Full Course Intro.
Angle between vectors leads to defining the Dot Product | Multivariable Calculus.
Geometrically Defining the Cross Product | Multivariable Calculus.
The Vector Equation of Lines | Multivariable Calculus.
Equations of Planes: Vector & Component Forms | Multivariable Calculus.
3D Curves and their Tangents | Intro to Vector-Valued Functions.
How long is a curve?? The Arclength Formula in 3D.
How curvy is a curve? Intro to Curvature & Circles of Curvature | Multi-variable Calculus.
Torsion: How curves twist in space, and the TNB or Frenet Frame.
Tangential and Normal components of Acceleration | Multi-variable Calculus.
Visualizing Multi-variable Functions with Contour Plots.
Limits are...weird...for multi-variable functions | Limits along paths.
Computing Multivariable Limits Algebraically.
What are derivatives in 3D? Intro to Partial Derivatives.
Continuity vs Partial Derivatives vs Differentiability | My Favorite Multivariable Function.
What is differentiability for multivariable functions??.
The Multi-Variable Chain Rule: Derivatives of Compositions.
Directional Derivatives | What's the slope in any direction?.
Geometric Meaning of the Gradient Vector.
How to find the TANGENT PLANE | Linear approximation of multi-variable functions.
Multi-variable Optimization & the Second Derivative Test.
Multivariable Optimization with Boundaries.
Lagrange Multipliers | Geometric Meaning & Full Example.
Lagrange Multipliers with TWO constraints | Multivariable Optimization.
Defining Double Integration with Riemann Sums | Volume under a Surface.
Double Integration Example over General Regions --- two ways!.
Change the order of integration to solve tricky integrals.
Double Integration in Polar Coordinates | Example & Derivation.
The Gaussian Integral // Solved Using Polar Coordinates.
Triple Integrals in Cartesian Coordinates | Volume between Surfaces.
Integration in Spherical Coordinates.
Change of Variables & The Jacobian | Multi-variable Integration.