MTH 254 Vector Calculus 1

Lecture Hours: 5
Credits: 5

Explores functions of many variables such as curves and surfaces in three-dimensional space, vectors, rates of change of functions of several variables, and optimization in multivariable models. Also explores multivariable integration with spherical and cylindrical coordinates. Offers the first of two courses in multivariable calculus.

Prerequisite: Placement into WR 115 (or higher), or completion of WR 090 (or higher); and completion of MTH 252 (or higher); 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 vectors, directional derivatives, gradient vectors, and multiple integrals.
Use inductive reasoning to develop mathematical conjectures involving vectors, differentials, partial derivatives, and double and triple integrals. Use deductive reasoning to verify and apply mathematical arguments involving these topics.
Use mathematical problem-solving techniques involving vectors, curvature, cylindrical and spherical coordinates, partial derivatives, curves, lines, planes and tangent planes, differentials, directional derivatives, gradients, and double and triple integrals.
Make mathematical connections and solve problems from other disciplines involving vectors, differentials, partial derivatives, and double and triple integrals.
Use oral and written skills to individually and collaboratively communicate about planes, lines, and curves in 3-space, the motion of particles in 3-space, and iterated integrals.
Use appropriate technology to enhance mathematical thinking and understanding; to sketch graphs in 2- and 3-space, and to solve mathematical problems involving planes, lines, curves, and surfaces in 3-space.

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

Three-Dimensional Analytic Geometry
- Coordinate systems
- Vectors
- Equations of lines and planes
- Quadric surfaces
- Vectors functions and space curves
- Arc length and curvature
- Motion
Partial Derivatives
- Functions of several variables
- Partial differentiation
- Tangent planes
- The differential
- Chain rule
- Extrema
- Lagrange multipliers
Multiple Integrals
- Double integrals
- Polar coordinates
- Triple integrals
- Cylindrical and spherical coordinates