Calculus IV

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)