Math Ladder · College

Calculus 3: Multivariable Calculus

Vectors and space curves, functions of several variables, partial derivatives, gradients, optimization and multiple integrals.

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45 topics in 5 units. Builds on Calculus 2: Integrals and Series. Leads to Calculus 4: Vector Calculus, Probability, Complex Analysis and Numerical Analysis. Not sure this is the right course? Find your rung.

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Unit 1 Vectors and Space

9 topics

Work with vectors, lines, planes and surfaces in three dimensions.

  1. Three-Dimensional Coordinates
  2. Vectors in Three Dimensions
  3. The Dot Product and Projections
  4. The Cross Product
  5. Lines in Space
  6. Planes
  7. Distances to Lines and Planes
  8. Cylinders and Quadric Surfaces
  9. Cylindrical and Spherical Coordinates

Unit 2 Vector-Valued Functions

5 topics

Describe curves in space and the motion along them.

  1. Vector-Valued Functions and Space Curves
  2. Derivatives and Integrals of Vector Functions
  3. Arc Length in Space
  4. Curvature and the Unit Tangent and Normal
  5. Velocity and Acceleration in Space

Unit 3 Partial Derivatives

12 topics

Study functions of several variables: limits, partial derivatives, the chain rule and gradients.

  1. Functions of Several Variables
  2. Functions of Three Variables and Level Surfaces
  3. Limits and Continuity
  4. Partial Derivatives
  5. Higher-Order Partials
  6. Tangent Planes and Linear Approximation
  7. Differentials and Error Estimates
  8. The Chain Rule for Several Variables
  9. Implicit Differentiation with Partials
  10. Directional Derivatives
  11. The Gradient
  12. Gradients, Level Sets and Normal Lines

Unit 4 Optimization

7 topics

Find and classify extreme values, with and without constraints.

  1. Local Extrema and the Second Derivative Test
  2. Absolute Extrema on Closed Regions
  3. Lagrange Multipliers
  4. Lagrange Multipliers with Two Constraints
  5. Gradient Descent
  6. Fitting a Model by Minimizing Loss
  7. Backpropagation Is the Chain Rule

Unit 5 Multiple Integrals

12 topics

Integrate over regions in the plane and in space, in any coordinates.

  1. Double Integrals over Rectangles
  2. Iterated Integrals and Fubini's Theorem
  3. Double Integrals over General Regions
  4. Reversing the Order of Integration
  5. Double Integrals in Polar Coordinates
  6. Mass, Center of Mass and Probability
  7. Surface Area of a Graph
  8. Triple Integrals
  9. Triple Integrals in Cylindrical Coordinates
  10. Triple Integrals in Spherical Coordinates
  11. Mass and Moments of Solids
  12. Change of Variables and the Jacobian
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