Calculus IV

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).