Unit 3 of 6

Differentiation of Functions of Several Variables

  • 11 lessons
  • Parts 25 to 46

Most quantities depend on more than one input. This unit studies functions of several variables through their graphs and level sets, defines limits and continuity in the plane and in space, and introduces partial derivatives.

The total derivative, tangent planes, the chain rule, the gradient and directional derivatives turn these into a full theory of local linear approximation. The implicit and inverse function theorems and second-order Taylor approximation follow, and the unit ends with local and absolute extrema and Lagrange multipliers.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Functions, Limits, and Partial Derivatives

  1. Functions of Several Variables

    How to read a function of two or three variables through its domain, range, graph, traces and level sets, and how to tell whether a domain is open, closed or bounded.

    Theory part 25Practice part 26

  2. Multivariable Limits and Continuity

    What it means for a function of several variables to have a limit, why two disagreeing paths disprove a limit while agreeing paths prove nothing, and how bounds by distance settle it.

    Theory part 27Practice part 28

  3. Partial Derivatives

    How to differentiate a function of several variables in one coordinate direction at a time, what those slopes mean on the graph, and when mixed second partials agree.

    Theory part 29Practice part 30

Derivatives and Approximation

  1. Total Derivative, Tangent Planes, and Differentials

    What it means for a function of several variables to be differentiable, how the Jacobian matrix, tangent plane and differential express it, and why partial derivatives alone do not guarantee it.

    Theory part 31Practice part 32

  2. The Multivariable Chain Rule

    How to differentiate a composition of multivariable functions by multiplying Jacobian matrices or summing over the routes of a dependency diagram, with every derivative evaluated at the right point.

    Theory part 33Practice part 34

  3. Directional Derivatives and the Gradient

    How to measure the rate of change of a function in any direction, why the gradient gives every such rate for differentiable functions, and how it points uphill and stands normal to level sets.

    Theory part 35Practice part 36

  4. Implicit and Inverse Function Theorems

    When an equation can be solved for some variables near a point, when a map has a local inverse, how to differentiate the solutions without finding them, and why the answers are only local.

    Theory part 37Practice part 38

  5. Multivariable Taylor Approximation

    How to build quadratic and higher Taylor models of a function of several variables with the Hessian and multi-index notation, and how to tell how far the model can be trusted.

    Theory part 39Practice part 40

Optimization

  1. Local Extrema and Saddle Points

    How to find the critical points of a function of several variables, classify them as local maxima, minima or saddles with the Hessian, and handle the cases where the test says nothing.

    Theory part 41Practice part 42

  2. Absolute Extrema on Regions

    When a function of two variables is sure to reach a largest and a smallest value on a region, and how to find them by checking the interior, every boundary piece and every corner.

    Theory part 43Practice part 44

  3. Lagrange Multipliers

    Why the gradient of a function is parallel to the constraint gradient at a constrained extremum, how to solve the resulting system, and which extra points and checks the method needs.

    Theory part 45Practice part 46