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

• use vector notation and the properties of vectors
• use vectors to describe various physical quantities (position, velocity, acceleration...)
• compute dot and cross-products and verify and use properties of these products
• determine angle/orientation between two vectors or one vector and standard basis vectors
• find scalar and vector projection of one vector onto another
• use vector operations to find area and volume defined by sets of vectors
• find vector, parametric or symmetric representations for an equation of a line in R³
• determine whether two lines are parallel, perpendicular or skew
• determine whether or not two lines intersect
• find vector or scalar equations for a plane
• determine and describe the orientation of two planes (angle between their normal vectors)
• determine the points (if any) of intersection between any two lines or planes
• determine the distance between a point and a line or plane
• identify and sketch the surface for a degree-two equation in three variables
• sketch regions bounded by two quadric surfaces
• work with cylindrical or spherical coordinate systems to describe points in R³
• use cylindrical or spherical coordinates to express curves or surfaces in R³
• find 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
• find unit tangent, principal normal vectors and tangent lines to space curves
• verify differentiation rules for vector functions
• 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
• 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
• establish and apply the chain rules
• find and interpret implicit partial derivatives
• find directional derivatives and gradients of functions
• find the maximum value of a directional derivative and interpret with respect to the gradient
• find and classify critical points of a function of two variables; solve associated optimisation problems
• use Method of Lagrange Multipliers to solve constrained optimisation problems
• set up double and triple Riemann sums over rectangular regions and convert notation to multiple integrals; evaluate the Riemann sums
• 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 to multiple integrals and evaluate
• change the representation of an integral from one set of coordinates to another and evaluate
• calculate the Jacobian of a transformation of coordinates to re-express integrals
• solve geometric and applied problems involving integration
• sketch vector fields on R²
• find the gradient vector field of a multi-variable function
• set up and evaluate line integrals with respect to arclength, or any of the independent variables; identify applications for line integrals and solve
• 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
• verify physical principles with the fundamental theorem of line integrals (conservation of energy...)