Calculus II

Quantities that depend on two or more inputs at once, their domains, their graphs as surfaces and their level curves read as a contour map, and why the calculus of one variable cannot simply be reused on them.

Coordinates and distance in three dimensions, vectors and their components, the dot product for angles and projections, the cross product for areas and normals, and the equations of lines and planes in space.

Limits in several variables defined through distance, the function $xy/(x^2+y^2)$ whose limit depends on the direction of approach, proofs by polar coordinates and the squeeze, and where continuous functions of several variables come from.

Slopes of a surface taken one coordinate direction at a time, computed with the rules of Calculus I, read as slopes of slices, taken twice to give second and mixed partials, and Clairaut's theorem on when the order of differentiation does not matter.

Differentiability defined by an error that shrinks faster than the distance, the tangent plane it guarantees, linear approximation checked against true values, and the total differential that propagates measurement error through a formula.

The rate of change of a composition derived from the tangent plane, first along a path and then with several independent variables, used to convert partial derivatives to polar coordinates and to turn implicit differentiation into the formula $dy/dx=-F_x/F_y$.

The directional derivative as the slope along any unit vector, the gradient that packages every such slope into one vector, the proof that it points uphill most steeply and crosses contours at right angles, tangent planes to level surfaces, and gradient descent.

Critical points where the gradient vanishes, the second derivative test derived from the second order Taylor polynomial, the saddle it detects and the case it cannot decide, the least squares line found as a minimum, and absolute extrema on a closed and bounded region.

Optimising on a constraint: why the gradients of the objective and the constraint must be parallel at the answer, the method as a system of equations worked on boxes, distances and a production budget, what the multiplier $\lambda$ measures, and two constraints at once.

Volume under a surface as a limit of sums over small rectangles, evaluated as two single integrals by Fubini's theorem, over rectangles and then over regions between two curves, reversing the order of integration to reach an integral that was impossible the other way, and area, average value and mass.

Polar coordinates and the area element $rdrd\theta$ derived from the area of a small polar rectangle, the Gaussian integral $\underset{-\infty }{\overset{\infty }{\int }}{e}^{-x^2}dx=\sqrt{\pi }$ they make possible, and the Jacobian determinant that measures how any substitution in two variables stretches area.

Sums over small boxes in space evaluated as three single integrals, cylindrical and spherical coordinates with their volume elements derived, the volume of a ball, and the mass, centre of mass and moment of inertia of a solid body.