Which direction makes a multivariable function increase fastest? Learn how the gradient packages partial derivatives, how to calculate directional derivatives with unit vectors and dot products, and why gradients cross level curves at right angles.
Master cylindrical and spherical coordinates through visual geometry, conversion formulas, Jacobians, bounds, and worked triple integrals. This Calculus 3 lesson compares both systems, solves an upper hemisphere in two ways, and shows how symmetry determines the best coordinates.
The Divergence Theorem converts outward flux through a closed surface into a triple integral of divergence over the enclosed solid. Learn the theorem’s conditions, flux and divergence meanings, method-selection rules, singularity warnings, and a complete sphere example with exact verification.
Why do gradients become parallel in constrained optimization? This Woody Calculus lesson explains the geometry and algebra of Lagrange multipliers, derives the system ∇f = λ∇g, and solves complete rectangle and box optimization examples.
Stokes’ Theorem says that boundary circulation equals the total curl through a surface. In this Woody Calculus visual lesson, learn how Stokes’ Theorem connects line integrals, surface integrals, curl, orientation, the right-hand rule, Green’s Theorem, and vector calculus in Calculus 3.
Green’s Theorem turns a boundary line integral into an interior double integral. In this Woody Calculus visual lesson, learn how Green’s Theorem connects circulation, curl, flux, area, line integrals, double integrals, Stokes’ Theorem, and the Divergence Theorem in Calculus 3.
The Jacobian is the hidden scale factor behind every coordinate change in Calculus III. In this Woody Calculus visual lesson, learn how Jacobians explain area and volume scaling, change of variables, polar coordinates, cylindrical coordinates, spherical coordinates, double integrals, triple integrals, and why the mysterious extra factors r and ρ²sinφ appear.
The Möbius strip is one of the clearest examples of why orientation matters in Calculus 3, vector calculus, topology, and surface integrals. Learn how one half-twist creates a one-sided surface with one boundary edge, no global normal vector, and a powerful obstruction to the standard global form of Stokes’ Theorem.
Line integrals are one of the core ideas in Calculus 3 and vector calculus. Learn what line integrals measure, how vector fields interact with paths, why direction matters, and when different paths from the same start to the same end can produce different work.