🗻Math 21a: Multivariable Calculus

Points, vectors, and the two products that measure everything between them: angles, areas, and volumes.

1.1 Coordinates and Distance in Three Dimensions · 1.2 Vectors: Magnitude and Direction · 1.3 Vector Addition and Scalar Multiplication · 1.4 The Dot Product · +10 more

Curves traced by a moving point, and surfaces sliced open by planes until every quadric shape gives up its structure.

2.1 Parametrized Curves · 2.2 Velocity and Acceleration Vectors · 2.3 Arc Length of a Space Curve · 2.4 The Unit Tangent Vector and Curvature · +10 more

Slicing a surface to turn slopes into single-variable calculus, then bundling every slope at a point into one arrow: the gradient.

3.1 Limits in Several Variables · 3.2 Continuity and the Path Test for Limits · 3.3 Partial Derivatives · 3.4 Higher-Order Partial Derivatives and Clairaut's Theorem · +9 more

Finding hilltops and mountain passes with the gradient, then fencing the search onto a curve where Lagrange's tangency trick takes over.

4.1 Critical Points of a Function of Two Variables · 4.2 The Second Derivative Test · 4.3 Saddle Points · 4.4 Global Extrema on Closed and Bounded Regions · +5 more

Volume under a bumpy surface, swept region by region, in whichever order and coordinate system makes the sweep easiest.

5.1 Double Integrals over Rectangles · 5.2 Iterated Integrals and Fubini's Theorem · 5.3 Double Integrals over General Regions · 5.4 Changing the Order of Integration · +9 more

Volume, mass, and rotational inertia swept through solid regions, in Cartesian, cylindrical, and spherical coordinates.

6.1 Triple Integrals over Boxes · 6.2 Triple Integrals over General Solid Regions · 6.3 Triple Integrals in Cylindrical Coordinates · 6.4 Triple Integrals in Spherical Coordinates · +6 more

An arrow at every point in the plane, and the integral that tracks how much a field pushes along a path through it.

7.1 Vector Fields · 7.2 Gradient Fields and Potential Functions · 7.3 Line Integrals of Scalar Functions · 7.4 Line Integrals of Vector Fields · +7 more

The first fundamental theorem of vector calculus: every interior rotation or flow cancels away until only the boundary is left.

8.1 Green's Theorem: Circulation Form · 8.2 Green's Theorem: Flux Form · 8.3 Computing Area with Green's Theorem · 8.4 Green's Theorem and Holes in the Domain · +2 more

A paddlewheel that spins or sits still, a point that sources or sinks fluid, and the surfaces that flux gets measured through.

9.1 The Curl of a Vector Field · 9.2 The Divergence of a Vector Field · 9.3 Parametrized Surfaces · 9.4 The Normal Vector to a Parametrized Surface · +5 more

Boundary versus interior, argued one dimension higher, until every theorem in the course turns out to be the same sentence.

10.1 Stokes' Theorem · 10.2 The Divergence Theorem · 10.3 Applications of the Divergence Theorem · 10.4 Choosing the Right Theorem · +4 more