Calculus 3
Multivariable calculus, answer first
Multivariable calculus is where one variable becomes several and most of the intuition from AP Calculus stops transferring. These pages give the answer up front, then the method, then the mistake the surface actually invites. 245 of them so far.
Every stated partial, gradient, integral value, classification and limit verdict here is recomputed numerically on every build, and the build fails if one stops checking out. This is not an AP course and carries no College Board alignment.
- Partial derivativesDifferentiate with respect to one variable and hold the rest still.70
- Second and mixed partialsSecond and mixed partials, and why the mixed ones come out equal.19
- GradientsThe vector of partials: the direction of steepest increase.34
- Directional derivativesThe rate of change in a chosen direction, not along an axis.12
- Double integralsVolume under a surface, and the order the bounds have to go in.34
- Triple integralsIntegrating over a solid region in three variables.10
- Critical points and classificationWhere both partials vanish, and whether it is a max, a min or a saddle.22
- Multivariable limitsWhether a limit in two variables exists, and the paths that decide it.24
- DivergenceHow much a vector field spreads out from a point.10
- CurlHow much a vector field rotates about a point.10
Coming from AP Calculus?
The single-variable material lives under AP Calculus BC. Partial derivatives are ordinary derivatives with the other variables held still, so the derivative rules all still apply, one variable at a time.