Mr. Integral

Think in more dimensions

Calculus III

Take calculus into space. Explore surfaces, gradients, multiple integrals and vector fields.

Start learning 6 units · 28 lessons

The course, unit by unit.

Open a unit for its introduction, map and lesson route.

01Vectors and Geometry in SpaceCoordinates, projections, planes and parametrized motion in three dimensions.4 lessons

How do we describe direction and motion in space?

Start with the introduction & map →
Unit cheat sheet All the key ideas in one reference
  1. 01Vectors, Dot Products and Cross Products
  2. 02Lines and Planes in Space
  3. 03Space Curves, Velocity and Acceleration
  4. 04Arc Length, Curvature and Turning
02Partial Derivatives and Local ChangeRead surfaces, test limits and connect partial derivatives to gradients.4 lessons

How does a surface change in different directions?

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Unit cheat sheet All the key ideas in one reference
  1. 01Surfaces, Level Sets and Multivariable Limits
  2. 02Partial Derivatives and Differentiability
  3. 03Gradients and Directional Derivatives
  4. 04Multivariable Chain Rules and Implicit Surfaces
03Optimization in Several VariablesClassify stationary points, check boundaries and apply constraints.4 lessons

Where is the best value when movement has limits?

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Unit cheat sheet All the key ideas in one reference
  1. 01Critical Points and the Hessian
  2. 02Absolute Extrema on Closed Regions
  3. 03Lagrange Multipliers and Constraint Geometry
  4. 04Multivariable Taylor Approximation and Error
04Double IntegralsAccumulate over regions, change integration order and use polar coordinates.4 lessons

How do we accumulate over an area?

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Unit cheat sheet All the key ideas in one reference
  1. 01Double Integrals and Accumulation
  2. 02Drawing Regions and Changing Integration Order
  3. 03Polar Coordinates in Double Integrals
  4. 04General Changes of Variables and Jacobians
05Triple IntegralsBuild volume bounds, select coordinates and compute mass and centroids.3 lessons

How do shape and density determine a solid’s totals?

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Unit cheat sheet All the key ideas in one reference
  1. 01Triple Integrals and Solid Bounds
  2. 02Cylindrical and Spherical Coordinates
  3. 03Density, Centers of Mass and Moments of Inertia
06Vector CalculusConnect circulation and flux to derivatives inside a region.9 lessons

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

Start with the introduction & map →
Unit cheat sheet All the key ideas in one reference
  1. 01Scalar and Vector Line Integrals
  2. 02Conservative Fields and Potentials
  3. 03Green’s Theorem
  4. 04Divergence and Curl
  5. 05Surface Area and Scalar Surface Integrals
  6. 06Oriented Surface Flux
  7. 07The Divergence Theorem
  8. 08Stokes’ Theorem
  9. 09Calculus III: Engineering Checkpoint