Variables are all around us: temperature, altitude, location, profit, color, and countless others. Multivariable Calculus is the tool of choice to shed light on complex relationships between 2, 3, or hundreds of variables simultaneously.
How does one control a robot whose motion depends on several variables at once?
How does an oceanographer understand carbon absorption of the ocean?
How can one assess if a prediction model matches data optimally?
How can one design policy to affect the behavior of consumers in order to better protect the planet?
All of these questions involve understanding vectors and derivatives of multivariable functions.
In this course, we begin our exploration of functions of several variables. We will start with learning to visualize multivariable functions, then move to computing and interpreting their derivatives. You will discover how to use linear approximations in several variables to simplify complex questions and will start to think about the world through multivariable dependencies.
Unit 1: Functions of two variables
Graphing functions and slicing
Unit 2: Geometry of derivatives
Unit 3: Optimization
Second derivative test
Least squares approximation
Unit 4: Matrices
Systems of equations
Undetermined and overdetermined systems
Linear functions as matrix maps
Unit 5: Curves
Parametric curves in 2 and 3 dimensions
Vectors and derivatives in three dimensions and higher
Larry Guth, Jennifer French, Duncan Levear and Denis Auroux