Calculus III

At the completion of the course a student will be expected to:

• Use and apply vector notation and the properties of vectors to describe various physical quantities
• Compute dot and cross-products and use the results to determine angle/orientation between two vectors or one vector and standard basis vectors
• Find scalar and vector projection of one vector onto another
• Find area and volume defined by sets of vectors
• Find vector, parametric or symmetric representations for equations of lines and planes in R³
• Determine whether two lines intersect, are parallel, perpendicular or skew
• Determine and describe the orientation of two planes using the angle between their normal vectors
• Determine the distance between a point and a line or plane, between two lines or between two planes.
• Identify and sketch quadric surfaces
• Use cylindrical or spherical coordinate systems to describe points, curves and surfaces in R³

• Evaluate limits involving vector functions
• Find the domain of a vector function and subsets of the domain where a vector function is continuous
• Sketch graphs of vector functions
• Differentiate and integrate vector functions, use differentiation rules for vector functions
• Find unit tangent, principal normal vectors and tangent lines to space curves
• Find the length of a space curve over an interval and its curvature 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, establish and apply chain rules, find and interpret implicit partial derivatives
• Find the equation of the tangent plane to a surface at a point
• Use differentials 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 and evaluate Riemann sums in polar coordinates and convert 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

(Time permitting)

• Sketch vector fields on R²
• Find the gradient vector field of a multi-variable function
• Evaluate line integrals for vector fields
• Determine whether or not a vector field is conservative
• Find conditions for and use the fundamental theorem of line integrals, apply the results