From y=f(x) to parametric equations, including self-intersection and point-by-point graph construction.
Tangent lines, second derivatives, concavity, and arc length for parametric curves.
Definition of polar coordinates and conversion formulas with Cartesian coordinates
Three-dimensional coordinates, implicit functions, distance formulas, and spheres
3D vectors, magnitude, standard basis vectors, and unit vectors
Definition, algebraic properties, angle formula, geometric meaning, and projections
Definition via determinant, geometric meaning, area formula, and scalar triple product
Parametric equations of lines in 3D, line segments as linear combinations, and equations of planes using normal vectors
Cylinders as extruded curves in 3D, and the six standard quadric surfaces obtained by reducing a general second-degree equation via completing the square
Vector-valued functions as maps from R to R^n, space curves, limits, continuity, and finding intersection curves
Derivative, tangent vectors, product rules, and component-wise integration for vector-valued functions
Arc length of 3D vector-valued functions, arc-length parameter, and an optional introduction to curvature and torsion
Scalar-valued functions of two variables, their domains and graphs, level sets, and standard 3D examples.
Epsilon-delta limits in one and two variables, path tests, continuity, and examples that show how to prove or disprove multivariable limits.
A refresher on single-variable derivatives, partial derivatives of functions of two variables, higher partials, mixed-partial symmetry, and geometric intuition.
The tangent line in one variable, tangent planes for functions of two variables, and how linear approximation leads to differentiability.
Chain rule for one- and two-parameter compositions, polar coordinates, implicit differentiation, and interactive examples.
Define the directional derivative via limits, compute it using partial derivatives, introduce the gradient vector, and explore how the gradient direction maximizes the rate of change.
Define local maxima and minima for functions of two variables, find critical points where the gradient vanishes, classify them with the second derivative test, and discuss absolute extrema on closed regions.
Optimize a function subject to a constraint g(x,y,z)=k by solving ∇f = λ∇g together with the constraint equation.
Build double integrals from 1D Riemann sums, define volume under a surface, and evaluate double integrals as iterated integrals on rectangular domains using Fubini's theorem.
Extend double integrals from rectangles to Type I and Type II regions bounded by curves, set up iterated integrals for each type, and explore linearity, comparison, and area as a special case.
Define polar rectangles, derive the polar area element r dr dθ, apply the polar change-of-variables formula to iterated integrals, and handle general polar domains.