Mr. Integral

THE BIG IDEA

Vector Calculus

How do local changes inside a field relate to its boundary?

Begin with accumulation and work along a curve. Then connect local rotation and expansion to circulation and flux, choosing a theorem only after checking the region, orientation and field assumptions.

Begin the first lesson →See the lesson route ↓

9 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 with01Path integrals
  2. derivatives reveal02Local field behavior
  3. crossing a boundary measures03Surface flux
  4. summing interior sources predicts04Integral theorems

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 unit9 lessons
  1. 01Scalar and Vector Line IntegralsDistinguish accumulation along a wire from work along an oriented path.
  2. 02Conservative Fields and PotentialsUse a potential to evaluate work and recognize when a curl test is insufficient.
  3. 03Green’s TheoremConvert planar circulation to a double integral with the correct boundary orientation.
  4. 04Divergence and CurlRead local expansion and rotation from derivatives of a vector field.
  5. 05Surface Area and Scalar Surface IntegralsUse tangent vectors to measure area and accumulate density on a curved surface.
  6. 06Oriented Surface FluxChoose a normal and calculate the amount of a vector field crossing a surface.
  7. 07The Divergence TheoremRelate outward flux through a closed boundary to sources inside a volume.
  8. 08Stokes’ TheoremMatch boundary orientation to a surface normal and integrate curl over a convenient surface.
  9. 09Calculus III: Engineering CheckpointCombine local approximation, constrained optimization and integral theorems in defensible models.