Mr. Integral

THE BIG IDEA

Partial Derivatives and Local Change

How does a surface change in different directions?

A multivariable function has many approach paths and many possible directions of motion. Move from level sets to coordinate rates, then combine those rates into a local model and directional predictions.

Begin the first lesson →See the lesson route ↓

4 lessons · No deadlines or locked lessons

SEE THE CONNECTIONS

How the ideas fit together.

A map of the main ideas. Follow a node to its lesson.

  1. Start with01Surfaces & limits
  2. varying one input measures02Coordinate rates
  3. assembling rates produces03The gradient
  4. changing several inputs requires04Dependency chains

YOUR LEARNING ROUTE

One idea at a time.

Read the explanation, use the visualization, then try the check before opening its reasoning.

Keep the unit cheat sheet handy →
Lessons in this unit4 lessons
  1. 01Surfaces, Level Sets and Multivariable LimitsUse domains and level curves to read a surface and distinguish a path test from an all-path proof.
  2. 02Partial Derivatives and DifferentiabilityCompute coordinate rates and determine when they combine into a valid local linear model.
  3. 03Gradients and Directional DerivativesNormalize a direction and use the gradient to predict local increase along it.
  4. 04Multivariable Chain Rules and Implicit SurfacesTrack every dependency through a composition and use a nonzero partial to solve locally for a variable.