Multivariable Calculus

Partial derivatives, gradients, multiple integrals, and the three great theorems of vector calculus.

Partial Derivatives — f / x: differentiate in x, hold y constant.
Multivariable Chain Rule — dz/dt = f_x dx/dt + f_y dy/dt — one term per path.
Gradient — f — direction of steepest ascent, perpendicular to level sets.
Lagrange Multipliers — Extremize f on g = c by solving f = λ g.
Double Integrals — _R f(x, y) dA = _a^b _c^d f dy dx — iterated integration, Fubini, general regions.
Triple Integrals — _V f dV — 3D iterated integration, with cylindrical (r) and spherical ( ^2 ) Jacobians.
Vector Fields (div, curl) — F ℝ^n → ℝ^n — divergence ( · F) measures outflow, curl ( × F) measures rotation.
Line Integrals — _C f ds (arc-length) and _C F · d r (work) — parameterize, dot, integrate.
Surface Integrals — _S f dS and _S F· d S — area weighted by f, or net flux of F through S.
Green's Theorem — _C (P dx + Q dy) = _D (Q_x - P_y) dA — circulation around the boundary equals total curl inside.
Stokes' Theorem — _C F· d r = _S ( × F)· d S — circulation around the boundary equals curl flux through any surface.
Divergence Theorem — _ V F· d S = _V ( · F) dV — outward flux equals total divergence inside.

Further Topics

Tangent Planes & Linear Approximation — The tangent plane is the linearization that approximates a surface.
Directional Derivatives — Rate of change along a unit vector via the gradient dot product.
Change of Variables & the Jacobian — Substitution in multiple integrals via the Jacobian determinant.
Parametric Surfaces & Surface Area — Parametrize r(u,v) and integrate | r_u × r_v| for surface area.
Conservative Fields & Potential Functions — Test for a potential, recover f, integrate by endpoints.
Limits & Continuity of Multivariable Functions — Path-dependence of limits and what continuity means in the plane.
3D Vectors & the Cross Product — One computation, two payoffs: a perpendicular direction and an area.
Arc Length & Curvature of Space Curves — Speed, the arc-length integral, and how sharply a space curve bends.