Calculus IV
Curriculum guideline
Lecture: 4 hrs/week
Tutorial: 1 hr/week
Lecture, tutorials, problems sessions and assignments
- Vector algebra: vector products, vector identities, tensor notation, use of vectors in geometry
- Vector-valued functions: differentiation, space curves and parameterizations, geometric properties of curves, physical interpretations of parameterizations
- Scalar and vector fields: gradient, divergence, curl, laplacian, cylindrical, spherical, orthogonal curvilinear coordinates, vector differential operator identities
- Integration: Line integrals, orientable surfaces and surface integrals, volume integrals
- Integral Theorems: Green’s theorem, the divergence theorem and Stokes’ theorem and their applications and consequences
- Potential Theory: Simply connected domain, conservative and solenoidal fields and their potentials
Upon completion of this course, successful students will be able 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.
- give a geometric interpretation of the scalar product, vector product, triple scalar product and triple vector product.
- use the geometric interpretation of the scalar product, vector product, triple scalar product and triple vector product in simplifying expressions or solving problems involving vectors.
- find the orientation of vectors via the right-hand rule.
- prove algebraic vector identities and simplify algebraic vector expressions with and without tensor notation.
- define the concepts of the limit and the derivative in the context of vector-valued functions.
- apply differentiation rules to vector-valued functions.
- describe 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 and basis vectors to solve problems involving space curves in R2.
- define a scalar field; give examples of scalar fields found in physical applications such as temperature, pressure, and concentration.
- determine the gradient of a scalar field; use the gradient to solve problems involving scalar fields.
- give a geometric interpretation of the gradient of a scalar field, the isotimic (level) surfaces of a scalar field, and the relationship between the two.
- define a vector field; give examples of vector fields found in physical applications such as gravitational fields, electric fields, and velocity fields.
- sketch and identify simple vector fields in R2.
- find the equations of flow lines for a given vector field.
- calculate the divergence and curl of a vector field.
- simplify expressions and solve problems involving gradient, divergence, and curl using del (nabla) notation.
- give a geometric interpretation of the divergence and curl of a vector field.
- define the concepts of incompressible (solenoidal) vector fields and irrotational vector fields.
- calculate the laplacian of scalar and vector fields.
- apply the concepts of gradient, divergence, curl, and laplacian to solve problems involving physical applications such as fluid flow, gravitation, and electricity and magnetism.
- verify vector differential 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 of scalar and vector fields along various curves, both open and closed.
- give a geometric interpretion of line integrals in terms of the circulation of a vector field.
- give a physical interpretation of line integrals in terms of work done by a vector field.
- solve problems involving line integrals arising from physical applications such as electricity and magnetism.
- determine if a line integral is path independent.
- use Green's theorem to simplify the evaluation of line integrals in the plane.
- determine if a region is a domain and, if so, whether it is simply connected.
- utilize the concept of an irrotational vector field and path independence of integrals to determine if a vector field is conservative.
- find a scalar potential function for a conservative vector field.
- determine if a vector field is incompressible (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.
- calculate surface integrals of scalar and vector fields over various surfaces, both open and closed.
- give an interpretation of surface integrals in terms of the flux of a vector field through a surface.
- solve problems involving surface integrals arising from physical applications such as electricity and magnetism and fluid flow.
- use Stokes' theorem to simplify the evaluation of surface integrals.
- compute a given volume integral.
- use cylindrical and spherical coordinates and basis vectors to evaluate surface and volume integrals.
- utilize the divergence theorem to evaluate given integrals.
- interpret the meaning of the divergence theorem in terms of the flux of a vector field.
- use Green’s theorem to find particular areas and evaluate given line integrals.
- utilise Stokes’ theorem to evaluate given integrals.
- interpret the meaning of Stokes’ theorem in terms of the circulation of a vector field.
- verify the flux and the Reynolds transport theorems (optional).
- prove various statements using the Fundamental Theorem of Vector Analysis (optional).
- use vector differential operator identities and the divergence theorem to derive Green's identities (optional).
Evaluation will be carried out in accordance with the Douglas College Evaluation Policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on 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% |
Consult the Douglas College bookstore for the latest required textbooks and materials.
Example textbooks and materials may 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.