MTH 255 Vector Calculus 2

Lecture Hours: 4
Credits: 4

Explores vector fields, motion in space, Green’s Theorem, Stokes’ Theorem, the Divergence Theorem, surface areas, and line and surface integrals along with their related topics, including divergence, curl, and flux. Offers the second course in multivariable calculus.

Prerequisite: Placement into WR 115 (or higher), or completion of WR 090 (or higher); and completion of MTH 254 (or higher) or equivalent course as determined by instructor; or consent of instructor. (All prerequisite courses must be completed with a grade of C or better.)
Student Learning Outcomes:

Create mathematical models of abstract and real-world situations using vector fields, line integrals, and surface integrals.
Use inductive reasoning to develop mathematical conjectures involving vector fields, line integrals, and surface integrals. Use deductive reasoning to verify and apply mathematical arguments involving these topics.
Use mathematical problem-solving techniques involving vector fields, line integrals, the fundamental theorem of line integrals in conservative fields, curl, divergence, flux, surface areas, surface integrals, Green’s Theorem, Stokes, Theorem, and the Divergence Theorem.
Make mathematical connections and solve problems from other disciplines involving vector fields, line integrals, and surface integrals.
Use oral and written skills to individually and collaboratively communicate about vector fields, line integrals, and surface areas and integrals.
Use appropriate technology to enhance mathematical thinking and understanding; to sketch vector fields and surfaces in 3-space, and to solve mathematical problems involving vector fields, line integrals, and surface areas and integrals.
Do projects that encourage independent, nontrivial exploration of vector fields, line integrals, and surface areas and integrals, and of some of the proofs at the heart of those topics.

Statewide General Education Outcomes:

Use appropriate mathematics to solve problems.
Recognize which mathematical concepts are applicable to a scenario, apply appropriate mathematics and technology in its analysis, and then accurately interpret, validate, and communicate the results.

Content Outline

Vector Fields
Line Integrals
- Curves
- Evaluating line integrals
- Work and energy
- Conservative fields
- Energy
- Green’s Theorem
- First and second moments
- Curl
- Divergence
- Vector forms of Green’s Theorem
Surfaces
- Areas
- Surface integrals
- Surface integrals of vector fields
- Flux
- Fluid and heat flow
- Electric field flux
- Curl
- Stokes’ Theorem
- Circulation
- Divergence (Gauss’) theorem
- Electromagnetic fields
Jacobian