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Several Variables Calculus

This is an overview of several variable calculus.


  • You should visit the sections below in order starting with Section 1.
  • Each section below has several videos explaining the listed concepts, exercises with solution videos, and self-assessment questions.
  • If you have watched the videos below and want to see more videos to better understand the material, you can see Chapter 14 of our course MATH 251: Several Variables Calculus. Note these MATH 251 videos are outside the scope of this mini-course and not required viewing. 

Several Variables Calculus

View Section 1: Functions of Several Variables
  • ​The definition of a function of two variables
  • The graph of a function of two variables with domain \(D\) and range \(R\)
  • The level curves of a function of two variables
View Section 2: Limits and Continuity
  • Calculating the limit of a surface
  • The definition of the limit of a two-variable function
  • Limits at infinity and infinite limits of two-variable functions
View Section 3: Partial Derivatives
  • ​The definition the partial derivative of \(f(x,y)\) with respect to \(x\) and \(y\)
  • The geometric interpretation of the partial derivative
  • Higher order partial derivatives and Clairaut’s Theorem
View Section 4: Tangent Planes and Linear Approximations
  • ​The equation of the tangent plane
  • Differentials
  • Applications of differentials
View Section 5: The Chain Rule
  • The chain rule for functions of more than one variable
  • Related rates
View Section 6: Directional Derivatives and The Gradient Vector
  • The Directional Derivative
  • The gradient vector
View Section 7: Maximum and Minimum Values
  • Local and absolute extrema of a function \(z = f (x, y)\)
  • The Second Derivative Test for Local Extrema
  • Extreme Value Theorem for Functions of Two Variables
View Section 8: Lagrange's Theorem
  • Explanation of Lagrange's Theorem
  • Finding the absolute maximum or absolute minimum values of \(z=f(x,y)\) subject to a constraint \(g(x,y)=k\)