Topics

Plot points in 3D using the right-hand rule, compute distance and midpoint, recognize spheres, and describe surfaces and regions with equations and inequalities.

Represent vectors as arrows or components, compute magnitudes and unit vectors, add and scale them, and apply them to forces, tensions, and velocities.

Compute dot products component-wise and geometrically; find angles, test orthogonality, project, and compute work done by a force.

Compute the cross product via determinant or component formula, use the right-hand rule for direction, and apply it to areas, normal vectors, and parallelism tests.

Write parametric and symmetric equations for lines, equations for planes via point-and-normal, find intersections, distances, and classify pairs as parallel, orthogonal, or skew.

Parameterize curves in 2D and 3D, find domains and limits of vector functions, eliminate parameters, and use coordinate-plane projections to visualize 3D curves.

Differentiate and integrate vector-valued functions component by component. Compute the unit tangent vector, tangent line, position–velocity–speed, and arc length; reparameterize by arc length.

Determine domains, sketch surfaces $z = f(x,y)$, and visualize them with traces, level curves, and contour maps.

Evaluate limits of functions of several variables by substitution, prove non-existence using the two-path method, and characterize continuity.

Compute partial derivatives, mixed partials, and use the chain rule to handle compositions involving several variables.

Compute gradients and directional derivatives. Find tangent planes and normal lines to surfaces. Use linearization and differentials for approximation.

Find local maxima, minima, and saddle points of $f(x, y)$ using critical points and the second derivative test. Find absolute extrema on closed bounded regions.

Set up and evaluate double and triple integrals as iterated integrals; find area, volume, mass, and average value; switch the order of integration.

Convert between rectangular and polar/cylindrical/spherical coordinates and use the right system to simplify integrals.

Sketch vector fields, compute divergence ($\nabla \cdot \mathbf F$) and curl ($\nabla \times \mathbf F$), test whether a field is conservative, and find potential functions.

Compute scalar and vector line integrals; recognize work, circulation, and flux setups; use the Fundamental Theorem of Line Integrals; apply Green's Theorem in circulation and flux forms.

Parameterize surfaces $\mathbf r(u, v)$, find a normal vector $\mathbf r_u \times \mathbf r_v$, and compute surface area for parametric and explicit surfaces.