Calculus III (Multivariable)
Calculus set loose in three dimensions: surfaces sliced into curves, gradients climbing hills, and volumes swept two different ways.
5 units · 125 topics · 593 questions
Points and vectors, and their two products, build the lines, planes, quadric surfaces, and curved coordinate grids every later chapter sits inside.
2.1 The Three-Dimensional Coordinate System · 2.1 Vectors and Vector Algebra · 2.2 Vectors in Three Dimensions and the Standard Basis · 2.2 Magnitude and Unit Vectors · +19 more
A moving point's position becomes one vector-valued function, and its derivatives hand you velocity, curvature, torsion, and the frame riding the curve.
3.1 Vector-Valued Functions and Parametrizing a Curve in Space · 3.1 Recognizing Common Space Curves · 3.1 Limits and Continuity of Vector-Valued Functions · 3.2 Derivatives of Vector-Valued Functions · +15 more
A surface sliced by planes turns partial derivatives into ordinary derivatives you can watch, building to gradients, saddle points, and Lagrange multipliers.
4.1 Functions of Two Variables and Their Graphs · 4.1 Level Curves and Contour Maps · 4.1 Level Surfaces of Functions of Three Variables · 4.1 Traces: Slicing a Surface with a Plane · +27 more
A double integral is a volume under a surface, re-swept the other way, mapped through polar grids, then stretched by a Jacobian before tripling into solids.
5.1 Double Integrals over Rectangular Regions · 5.1 A Double Integral as the Volume Under a Surface · 5.1 Iterated Integrals and Fubini's Theorem · 5.2 Double Integrals over General Regions · +22 more
A vector field carries a dropped-in particle, curl spins a paddlewheel, and three theorems show the boundary already knows what the interior is doing.
6.1 Vector Fields and Their Flow Lines · 6.1 Gradient Fields and Potential Functions · 6.2 Line Integrals of a Scalar Function Along a Curve · 6.2 Line Integrals with Respect to Arc Length · +22 more