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Calculus III (Multivariable)

Free Calculus III (Multivariable) study guide: 8 units with summaries, key ideas, flashcards and 178 practice questions with step-by-step solutions.

Lines up with: GRE Mathematics Subject Test (calculus portion).

Units

1. UNIT 01

   Vectors & the geometry of space

   Multivariable calculus starts in 3D space. Vectors describe direction and size, and dot and cross products measure angles, areas and perpendicular directions.
2. UNIT 02

   Vector-valued functions

   A vector-valued function traces a curve in space. Differentiating it gives velocity; integrating gives position.
3. UNIT 03

   Functions of several variables

   Functions of several variables have a partial derivative for each input. Each one measures change in a single direction while the others are held fixed.
4. UNIT 04

   Gradient & optimization

   The gradient points in the direction of steepest increase. Setting it to zero finds critical points, and Lagrange multipliers handle constraints.
5. UNIT 05

   Double integrals

   Double integrals add up a function over a region in the plane, giving volume under a surface or total mass.
6. UNIT 06

   Triple integrals & change of variables

   Triple integrals extend to 3D regions. Cylindrical and spherical coordinates simplify regions with circular symmetry, using a Jacobian scale factor.
7. UNIT 07

   Vector calculus

   Vector fields assign a vector to every point. Line integrals add up a field along a curve, which in physics is work.
8. UNIT 08

   Surface integrals, Stokes & divergence

   Surface integrals measure flux through a surface. Stokes' and the divergence theorem connect them to line and volume integrals.