Multivariable Calculus

Determinants: Understanding Their Significance in Matrices

From Variables to Matrices in Mathematics and Machine Learning

Under Over-Determined Systems

Exploring Linear Systems and Intersections of Planes

Inverse Matrices: Definition and Properties

Identity Matrix in Mathematics and Science

Caution: order matters. AB is not BA

Exploring Matrix Multiplication: From Theory to Practice

Matrices: Understanding the Basics and Applications

Vector Components: Analyzing the General Case

Vector Rotation: Techniques and Formulas

Solving Chain Rule Problems with Multiple Variables

Differentials. Chain Rule Problem

Synthesis: Q and A

Chain Rule Expansion: Handling More Variables

Theoretical Application. Product Rule From Chain Rule

Justify the Chain Rule

Multivariable Chain Rule. Dividing Differentials

Robot Arm Matrix Example 2

Robot Arm Analysis: Linear Approximation

Solving the Robot Arm Controller Problem with Matrices

Recitation Video: Lagrange Multipliers with 3 Variables

Partial Derivatives and the Gradient. Gradient and Theorem

Lagrange Multiplier Method: Maximal Perimeter Rectangles

Maximum along the Boundary Continued: Lagrange Multipliers

Maximum along the Boundary Intuition. Setting up the Statement

Optimization and Linear Approximation

Where is the Maximum?

Increasing or Decreasing as You Move Clockwise?

Deciphering Bounded Regions: A Constrained Optimization Review

Optimization Intuition: Worked Example Continued

Constrained Optimization: Big Picture

Understanding Bounded Regions in Mathematical Optimization

Second Derivative Test. Worked Example Recitation

Second Derivative Test. Worked Example Conclusion

Comparison to the Quadratic Formula (1)

Critical Point Type for a Quadratic Function

Higher Order Partial Derivatives

Critical Point Practice. Worked Examples

Completed Worked Example

Critical Points Definition

Tangent Plane and Partial Derivatives: A Closer Look

Geometry Proofs using Coordinate Free Vectors

Practice in Higher Dimensions

What are Directional Derivatives

Function Gradients and Level Curves: Insights and Applications

Maximize a Linear Function

Visualizing the Gradient Vector

The Gradient Vector

Warm Up: Linear Approximations

Parallel Lines: Definition and Properties

Perpendicular Vectors

Vector Addition and Subtraction: Formulas and Examples

Differentiating Unit Vectors and Scalars

Magnitude: Short and Clear

Vector Examples in 2 Dimensions

Vector Introduction

Functions of Two Variables. Worked Examples

Linear Approximations and Derivatives in Multivariable Functions

Linear Approximations and Tangent Planes

Visualizing Linear Approximations and Tangent Planes in 3D

Partial Derivatives and Computing Techniques

Exploring Partial Derivatives and Their Definitions

Visualizing Functions: Contour Plots and Level Curves