Calculus III

Upon successful completion of the course, students will be able to:

• use and apply vector notation and the properties of vectors to describe various physical quantities;
• use dot and cross-products to solve various geometric problems involving vectors, points, lines, and planes;
• find vector, parametric, or symmetric representations for equations of lines and planes in R³;
• identify and sketch quadric surfaces;
• use cylindrical or spherical coordinate systems to describe points, curves and surfaces in R³;
• find the domain of a vector-valued functions of a single variable and subsets of the domain where vector-valued functions of a single variable are continuous;
• sketch graphs of vector-valued functions of a single variable;
• differentiate and integrate vector-valued functions of a single variable, and use differentiation rules for vector-valued functions of a single variable;
• find unit tangent, principal normal vectors and tangent lines to space curves;
• find the length of a space curve over an interval;
• find the curvature of a space curve at a point;
• apply the ideas of tangent and normal vectors and curvature to motion in space;
• sketch level curves for functions of two variables and level surfaces for functions of three variables;
• calculate limits (or prove the non-existence) for functions of two or three variables;
• find subsets of a function’s domain for which the function is continuous;
• calculate partial derivatives of a function, and establish and apply differentiation rules;
• find and interpret implicit partial derivatives;
• find the equation of the tangent plane to a surface at a point;
• use differentials or linear approximation to approximate values and errors for a function of two or three variables;
• find directional derivatives and gradients of functions;
• find and classify critical points of a function of two variables; solve associated optimization problems;
• use the Method of Lagrange Multipliers to solve constrained optimization problems;
• set up and evaluate double and triple Riemann sums over rectangular regions and convert notation to multiple integrals;
• identify different classes of domains of integration to set up and evaluate general multiple integrals;
• change the order of integration variables;
• set up Riemann sums in polar coordinates and convert them to multiple integrals;
• change the representation of an integral from one set of coordinates to another;
• calculate the Jacobian of a transformation of coordinates to re-express integrals;
• solve geometric and applied problems involving integration.