Multivariable Calculus Study Guide: 49 Flashcards, 3 Fixes and 4 Missing Formulas

Multivariable calculus is where you stop thinking of the derivative as a number and start thinking of it as a linear map. The Multivariable Calculus deck (code RH7346, 49 cards) is condensed from the encyclopedia article on the subject. As a vocabulary index it is excellent. As exam preparation it is thin: none of its 49 cards asks you to compute anything. Here is how to use it, and what to add.

Core cards from the deck

Also covered: limits along a path, directional derivatives D_v f, partial derivatives, the del operator, divergence and curl, multiple integrals, Fubini's theorem, line and surface integrals, the gradient theorem, the generalized Stokes theorem, Lipschitz continuity, arc length ∫|γ'(t)| dt, Lagrange multipliers, level sets, curvature κ = |dT/ds|, flux, scalar vs vector fields, the second-order Taylor expansion, critical points and saddle points.

Three cards to correct

• Card 23 (Fubini): "if the integrand is continuous" only works on a bounded rectangle. The general condition is absolute integrability (∬|f| < ∞), and for non-negative functions Tonelli's theorem drops even that requirement.
• Card 43 (saddle point): a Hessian with both positive and negative eigenvalues is sufficient for a saddle, but it is not part of the definition. For f = x² − y⁴, the Hessian at the origin is semidefinite and the point is still a saddle.
• Card 42 (critical point): add "or where the gradient is undefined". Otherwise |x| + |y| at the origin slips through.

The four cards you should write yourself

1. Second-derivative test: D = f_xx·f_yy − f_xy². If D > 0 and f_xx > 0, it's a minimum. If D > 0 and f_xx < 0, a maximum. If D < 0, a saddle. If D = 0, the test is inconclusive.
2. Lagrange condition: ∇f = λ∇g with g = c. The deck only says what Lagrange multipliers are used for (card 33).
3. Multivariable chain rule: dz/dt = f_x·x'(t) + f_y·y'(t). In matrix form, it's a product of Jacobians.
4. Change of variables: dA = r dr dθ (polar coordinates) and dV = ρ² sin φ dρ dθ dφ (spherical coordinates). These come from the Jacobian determinant.

How to study it

• Treat the four integral theorems as a single card family. On every card, ask "dimension of the region → dimension of its boundary". The gradient theorem goes from curve to endpoints, Green and Stokes from surface to curve, and the divergence theorem from solid to surface. They are all the generalized Stokes theorem ∫Ω dω = ∫∂Ω ω (card 30).
• Suspend cards 44-49 (applications in economics, optimal control and regression) until the end. They are trivia, not tools.

This deck builds on the one-variable pathologies from Limits & Continuity and the techniques from Integration. Review those first if path-dependent limits feel slippery.

Find RH7346 in the Flip It library and pair it with a problem set.