Calculus IV

Curriculum guideline

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Course
MATH 2440
Discontinued
No
Course code
MATH 2440
Descriptive
Calculus IV
Department
Mathematics
Faculty
Science & Technology
Credits
3.00
Start date
End term
201430
PLAR
No
Semester length
15
Max class size
35
Contact hours
Lectures: 4 hrs/week Tutorials: 1 hr/week
Method(s) of instruction
Lecture
Tutorial
Learning activities

Lecture, problems sessions, written and computer exercises.

Course description
This is a course in vector calculus that applies calculus to vector functions of a single variable as well as scalar and vector fields. Topics include gradient, divergence, curl; line, surface and volume integrals; the divergence theorem as well as the theorems of Green and Stokes.
Course content
  1. Review of vector algebra, scalar and vector fields; tensor notation, acceleration, curvature, Frenet formulae.
  2. Scalar and vector fields; gradient, divergence, curl, laplacian; cylindrical, spherical, orthogonal curvilinear coordinates.
  3. Line, surface and volume integrals; simply connected domain; conservative and solenoidal fields and their potential; orientable surfaces and surface integrals; volume integrals.
  4. The divergence theorem and Stokes’ theorem and their applications and consequences; the Fundamental Theorem of Vector Analysis and Green’s theorem.

Optional: Transport theorems.

Learning outcomes

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

  • Perform basic vector operations such as addition, subtraction, multiplication by a scalar, as well as find the magnitude of a vector
  • Find a unit vector in the same direction as a given vector
  • Use vectors to solve geometric problems and problems involving lines and planes in R3
  • Use, and understand the geometric significance of, the scalar, vector, and triple scalar products in problem solving
  • Find the orientation of vectors via the right-hand rule
  • Prove various vector identities
  • Use tensor notation to simplify vector expressions
  • Apply the concepts of the limit and differentiation to vector-valued functions
  • Reparametrize space curves, especially in terms of arc length; find the unit tangent vector to a given space curve
  • Find the velocity and (the tangential and normal components of) the acceleration of a particle moving along a space curve; find the curvature and torsion of a space curve
  • Apply polar, cylindrical and spherical coordinates to solve problems involving space curves
  • Determine, and solve problems using, the gradient of a scalar field; interpret the practical significance of the gradient of a scalar field and isotimic (level) surfaces
  • Find the equations of flow lines for a given vector field
  • Calculate and interpret geometrically the divergence and curl of vector fields; represent gradient, divergence, and curl using del (nabla) notation
  • Calculate the laplacian of scalar and vector fields
  • Verify vector operator identities with and without tensor notation
  • Compute grad, div, curl and laplacian in cylindrical, spherical and general orthogonal curvilinear coordinates
  • Calculate line integrals; interpret them especially in terms of work done
  • Determine if a region is a domain and, if so, whether it is simply connected
  • Utilize the concept of an irrotational vector field to determine if the field is conservative; find a potential function for a conservative vector field
  • Determine if a vector field is solenoidal and, if so, find a corresponding vector potential in simple cases
  • Construct a parametric representation of a surface and find the unit normal to the surface either parametrically or nonparametrically
  • Compute a given surface integral directly; give an interpretation for the surface integral
  • Compute a given volume integral
  • Utilize the divergence theorem to evaluate given integrals; interpret the practical meaning of the divergence theorem
  • Prove various statements involving Green’s formulae and the Fundamental Theorem of Vector Analysis
  • Use Green’s theorem to find particular areas and evaluate given line integrals
  • Utilise Stokes’ theorem to evaluate given integrals; interpret the practical meaning of Stokes’ theorem
  • Optional:  Use dyadics to compute Taylor polynomials, verify the flux and Reynold’s transport theorems
Means of assessment
Quizzes 0-40%
Term tests 20-70%
Assignments 0-20%
Attendance 0-5%
Participation 0-5%
Tutorial activities 0-10%
Final exam 30-40%

 

Textbook materials

Textbooks and Materials to be Purchased by Students

Davis and Snider. Introduction to Vector Analysis, Seventh Edition, Hawkes, 1995.

Prerequisites

MATH 2321

Corequisites

MATH 2232 (recommended)