# Calculus IV

## Curriculum Guideline

Effective Date:
Course
Discontinued
No
Course Code
MATH 2440
Descriptive
Calculus IV
Department
Mathematics
Faculty
Science & Technology
Credits
3.00
Start Date
End Term
Not Specified
PLAR
No
Semester Length
15
Max Class Size
35
Contact Hours
Lectures: 4 hrs/week Tutorials: 1 hr/week
Method Of Instruction
Lecture
Tutorial
Methods Of Instruction

Lecture, tutorials, problems sessions and assignments

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. Vector algebra: vector products, vector identities; tensor notation; use of vectors in geometry
2. Vector-valued functions: differentiation, space curves and parameterizations, geometric properties of curves, physical interpretations of parameterizations
3. Scalar and vector fields: gradient, divergence, curl, laplacian; cylindrical, spherical, orthogonal curvilinear coordinates; vector operator identities
4. Integration: Line integrals; orientable surfaces and surface integrals; volume integrals
5. Integral Theorems: Green’s theorem, the divergence theorem and Stokes’ theorem and their applications and consequences
6. Potential Theory: Simply connected domain; conservative and solenoidal fields and their potentials; Green’s formulas
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 product, vector product, triple scalar product and triple vector product in problem solving
• Find the orientation of vectors via the right-hand rule
• Prove algebraic vector identities and simplify algebraic vector expressions with and without tensor notation
• Apply the concepts of the limit and differentiation to vector-valued functions; apply differentiation rules to vector-valued functions
• Understand the difference between a space curve and a parameterization. Give parameterizations for common curves including, but not limited to, lines, circles and helixes
• Calculate geometric quantities related to space curves such as arc length, unit tangent vector, unit normal vector, curvature and torsion
• Reparametrize space curves, especially in terms of arc length
• Find the velocity, speed and acceleration (including tangential and normal components) of a particle moving along a space curve
• Apply polar coordinates to solve problems involving space curves in R2
• 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
• Sketch and identify simple vector fields in R2
• 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
• Represent vector fields and compute gradient, divergence, 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
• Use cylindrical and spherical coordinates to evaluate surface and volume integrals
• Utilize the divergence theorem to evaluate given integrals; interpret the practical meaning of the divergence theorem
• 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
• Prove various statements involving Green’s formulas
• Use dyadics to compute Taylor polynomials (optional)
• Verify the flux and the Reynolds transport theorems (optional)
• Prove various statements using the Fundamental Theorem of Vector Analysis (optional)
Means of Assessment

Evaluation will be carried out in accordance with Douglas College policy.  The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester.  Evaluation will be based on some of the following:

 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

Consult the Douglas College bookstore for the current textbook. Examples of textbooks under consideration include:

Introduction to Vector Analysis, Davis and Snider, Hawkes Publishing, Seventh Edition
Vector Calculus, Lovric, Wiley, current edition.
Vector Calculus, Marsden and Tromba, Freeman, current edition,
Vector Calculus, Colley, Pearson, current edition,

Prerequisites
Corequisites

MATH 2232 (recommended)