$\nabla$Multivariable Calculus

Lines and Planes

A line needs a point and a direction; a plane needs a point and a perpendicular. Write both, find where they meet, measure distances and angles in space, and meet skew lines, which never meet and are not parallel.

Curves and Motion

Follow a moving point: velocity, speed and acceleration as vectors. Measure how sharply a curve bends with the circle that hugs it best, and split acceleration into the part that speeds you up and the part that turns you.

Partial Derivatives

A hill has a different slope in every direction. Slice it to get partial derivatives, lay a tangent plane on it to approximate, read a contour map, and find the direction of steepest climb: the gradient.

The Multivariable Chain Rule

Walk along a path across a landscape: how fast does your height change? Add up the contributions from every route through the variables, draw the dependency tree, and differentiate curves you cannot solve for.

Optimization

Find the peaks, pits and passes of a landscape where the gradient is zero, tell them apart with second derivatives, and use Lagrange's brilliant idea for the best point on a constraint: where the gradient meets the curve at right angles.

Double Integrals

Volume under a surface, built from prisms and then from slices. Integrate one variable at a time, describe a region from two directions (and pick the easy one), and switch to polar coordinates for anything round.

Triple Integrals

Add up something spread through a solid: volume, mass, average temperature. Stack slices in the right order, then switch to cylindrical and spherical coordinates and finally see where 4/3 π r³ comes from.

Change of Variables

Swap coordinates to make an integral easy, but pay the exchange rate: the Jacobian, which measures how a tiny square gets stretched. See it for linear maps and polar coordinates, and choose substitutions that straighten out awkward regions.

Vector Fields and Green's Theorem

Wind maps and currents are vector fields. Measure how much a field pushes you around a loop (circulation) and how much flows out (flux), meet fields with potentials where loops always give zero, and watch Green's theorem turn a loop integral into an area integral.

Stokes and the Divergence Theorem

The grand finale of calculus: what happens inside a region is recorded on its boundary. Measure flux out of a loop, meet divergence (sources and sinks), and see Green, Stokes and Gauss as one theorem in three costumes.