topics

Geometry of space; vectors

    • Use Cartesian, cylindrical, and spherical coordinates, and be able to convert between the different coordinate systems.
    • Resolve geometric vectors into components parallel to coordinate axes.
    • Perform the operations of vector addition and scalar multiplication, and interpret them geometrically.
    • Use the dot product to calculate magnitude of a vector, angle between vectors, and projection of one vector on another.
    • Use the cross product; interpret the cross product geometrically and as area of a parallelogram.
    • Use and interpret geometrically the standard equation for a plane, and the parametric and vector forms for a line.
    • Use parametric equations for curves; find tangent lines to parametric curves; find length of parametric curves.
    • Compute velocity, unit tangent and acceleration vectors along a parametric curve.
    • Resolve acceleration into tangential and normal components; compute curvature.
    • Recognize cylinders and quadric surfaces from their Cartesian equations.

Differentiation for functions of two or more variables

    • Represent a function of two variables as the graph of a surface; sketch level curves; level surfaces.
    • Calculate partial derivatives. Find tangent planes to functions of two variables.
    • Use the Chain Rule; implicit differentiation of multi-variable functions.
    • Find the gradient and interpret the gradient geometrically. Use the gradient to find tangent planes.
    • Compute directional derivatives.
    • Use linear approximations to estimate the function near a point of tangency.
    • Find second-order Taylor polynomials for multi-variable functions.
    • Find and classify critical points of functions, using the second partials test.
    • Find maximum and minimum values for a function defined on a closed and bounded set.
    • Use the method of Lagrange multipliers in optimization.

Integration for multi-variable functions and vector calculus

    • Use iterated integrals to evaluate double integrals over regions in the plane.
    • Set up and evaluate double integrals in polar coordinates.
    • Set up and evaluate triple integrals over regions in space using Cartesian coordinates.
    • Set up and evaluate integrals to compute area, surface area, volume, mass, moments, center of mass, and inertia.
    • Set up and evaluate integrals in cylindrical and spherical coordinates.
    • Change the order of integration in multiple integrals.
    • Carry out change of variables (Jacobian) in multiple integrals.
    • Set up and evaluate line integrals of scalar functions or vector fields along curves. Compute work along a line.
    • Determine the curl and divergence of a vector field.
    • Recognize conservative vector fields, and apply the fundamental theorem for line integrals of conservative vector fields.
    • State and apply Green's Theorem.
    • Set up and evaluate surface integrals; compute the flux of a vector field through a boundary.
    • (*)State and apply Stokes' Theorem.
    • (*)State and apply the Divergence Theorem.

(*) = Topic covered during dead week; not on Exam 3