Math 214 - Multivariable Calculus - Schedule

Motion

I can draw parametric curves in 2D and 3D.
For a given parametrization, I can compute velocity, acceleration, speed, and give a vector equation of the tangent line to the curve at a given point.
I can use the dot product to compute angles and decompose a vector into parallel and orthogonal components.
I can use integrals to find the length of a parametric curve and the work done by a nonconstant force along a parametric curve.
I can compute the TNB frame for a space curve and find the curvature of a vector-valued function at a given point.

I can construct contour plots, surface plots, and gradient field plots for functions of the form $f(x, y)$, and I can construct level surface plots for functions of the form $f(x, y, z)$
For functions of several variables, I can compute partial derivatives, directional derivatives, tolerances (using differentials), and equations of tangent planes (linearizations).
I can obtain and use appropriate chain rules to compute derivatives for compositions of functions.
I can use Lagrange multipliers to locate and compute extreme values of a function $f$ subject to a constraint $g = c$.
I can apply the second derivative test, using eigenvalues, to locate local maximum and local minimum values of a function $f$ over a region $R$.

I can set up and compute iterated single, double, and triple integrals to obtain lengths, areas, and volumes, connecting these to the differentials $dx$, $ds$, $dA$, and $dV$.
I can find the average value of a function over a region, and use this to compute the mass, center-of-mass (varying density), and centroid (uniform density) of a wire, planar region, or solid object.
I can draw regions described by the bounds of an integral, and then use this drawing to swap the order of integration.
I can appropriately use polar coordinates $dA = |r| dr d\theta$, cylindrical coordinates $dV = |r|drd\theta dz$, and spherical coordinates $dV = |\rho ^2 \sin\phi |d\rho d\phi d\theta$ to setup and compute iterated integrals.
For a given change-of-coordinates, I can compute and appropriately use the Jacobian to change an integral from one coordinate system to another.

I can determine whether or not a vector field has a potential, and verify the Fundamental Theorem of Line Integrals for vector fields that have a potential (computing the work done by a vector field along a curve).
I can verify Green's Theorem given a simple closed curve and vector field (computing the circulation of the vector field along the curve).
For a parametric surface, I can draw the surface, as well as set up appropriate integrals to compute the surface area, mass, and center-of-mass.
I can compute the curl of a vector field and use it to verify Stokes's Theorem for a given parametric surface and vector field.
I can compute the divergence of a vector field and use it to verify the Divergence Theorem for a given closed surface and vector field, obtaining the flux of a vector field across the surface.