Calculus III

Curriculum Guideline

Effective Date:
Course Code
MATH 2321
Calculus III
Science & Technology
Start Date
End Term
Semester Length
15 weeks
Max Class Size
Contact Hours
Lectures: 4 hrs/week Tutorials: 1 hr/week
Method Of Instruction
Methods Of Instruction

Lecture, problems sessions, written and computer exercises.

Course Description
This course extends the theory of differential and integral calculus to functions of many variables. Topics include the study of vectors, quadric surfaces, vector functions, cylindrical and spherical coordinates, partial derivatives, multiple integrals, vector fields and line integrals; all with applications.
Course Content
  1. Properties and applications of points, curves and surfaces for various coordinates in R3.
  2. Operations, properties and applications of vectors and vector functions.
  3. Partial Derivatives: Limits, partial derivative rules and properties, gradients and optimization principles. Applications.
  4. Multiple Integrals: Double and triple integrals over general domains in appropriate coordinate systems (rectangular, polar, cylindrical, spherical or other defined coordinates). Applications.
  5. Vector Calculus: Vector fields, line integrals, Fundamental Theorem of Line Integrals. Applications. (If time permits)
Learning Outcomes

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...)
Means of Assessment


Term tests




Tutorial activities

Final examination








Textbook Materials

Stewart, James. Multivariable Calculus 5e, Brooks/Cole, 2003.



MATH 2232 is recommended

Which Prerequisite