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Calculus III

Course Code: MATH 2321
Faculty: Science & Technology
Department: Mathematics
Credits: 3.0
Semester: 15 weeks
Learning Format: Lecture, Tutorial
Typically Offered: Fall
course overview

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)

Methods of Instruction

Lecture, problems sessions, written and computer exercises.

Means of Assessment


Term tests




Tutorial activities

Final examination








Learning Outcomes

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

course prerequisites

MATH 1220 or MATH 1223


MATH 2232 is recommended

curriculum guidelines

Course Guidelines for previous years are viewable by selecting the version desired. If you took this course and do not see a listing for the starting semester/year of the course, consider the previous version as the applicable version.

course schedule and availability
course transferability

Below shows how this course and its credits transfer within the BC transfer system. 

A course is considered university-transferable (UT) if it transfers to at least one of the five research universities in British Columbia: University of British Columbia; University of British Columbia-Okanagan; Simon Fraser University; University of Victoria; and the University of Northern British Columbia.

For more information on transfer visit the BC Transfer Guide and BCCAT websites.


If your course prerequisites indicate that you need an assessment, please see our Assessment page for more information.