MATH 224 Multivariate Calculus (5 credits)
Distribution Area Fulfilled Natural Sciences; Quantitative and Symbolic Reasoning; General Transfer Elective
Prerequisite MATH& 153 with a grade of 2.0 or higher or instructor permission.
Course Description
Functions of several variables. Partial derivatives, multiple integrals, and their applications. Vector analysis including vector fields, line and surface integrals, Green’s theorem, Stokes’ theorem, and the Divergence theorem.
Course Content
A. Functions of several variables
B. Partial derivatives
C. Techniques of optimization including Lagrange multipliers
D. Double and triple integrals
E. Integration in polar, cylindrical, and spherical coordinates
F. Change of variables theorem for multiple integrals
G. Vector fields
H. Line and surface integrals of functions and vector fields
I. The gradient operator and the Fundamental Theorem of Line Integrals
J. The curl operator and Green’s Theorem
K. Stokes’ Theorem
L. The divergence operator and the Divergence theorem
M. Applications of partial derivatives, multiple integrals, and vector analysis to mathematics, the sciences, and engineering.
Student Outcomes
Functions of Several Variables component
1. Evaluate functions of several variables numerically, graphically, and symbolically.
2. Graph functions of several variables utilizing technology as appropriate.
3. Compute limits of functions of several variables.
4. Determine the domain and continuity of a function of several variables.
Partial Derivatives component
5. Utilize the definition of the partial derivative of a function of several variables to solve rate of change problems.
6. Compute partial derivatives symbolically utilizing the basic techniques from single variable calculus.
7. Determine an equation of the tangent place to a surface defined by a graph of a function, by an implicit equation, and by a parametric equation.
8. Compute partial derivatives via the chain rule and through implicit differentiation.
9. Locate and test extrema using the Second Derivative Test.
10. Utilize Lagrange multipliers to optimize functions of several variables given one or more constraints.
11. Apply directional derivatives to solve rate of change problems in arbitrary directions. Determine the direction of maximal and minimal change of a function of several variables.
12. Apply techniques of partial derivatives to solve problems in the sciences and engineering.
Multiple Integrals component
13. Construct the double integral of a function of two variables as the limit of a Riemann sum.
14. Compute double integrals of functions of two variables over rectangle regions by identifying the relevant solid and computing its volume.
15. Compute multiple integrals over general regions whose boundaries are parameterized by curves or surfaces, utilizing the basic techniques from single variable calculus.
16. Compute double and triple integrals in polar, cylindrical, and spherical coordinates.
17. Compute the surface area of a parameterized surface via a double integral.
18. Describe and construct basic transformations and compute their Jacobians.
19. Apply the Change of Variables theorem to compute multiple integrals.
20. Apply techniques of multiple integrals to solve problems in the sciences and engineering.
Vector Analysis component
21. Draw vector fields in two and three dimensions.
22. Qualitatively determine characteristics of vector fields, including periodic orbits and singularities.
23. Compute the line integral of a function and of a vector field along a parameterized curve.
24. Determine the work done by a force field in moving a mass-less particle along a parameterized trajectory.
25. Determine whether a vector field is conservative or not and construct a potential if one exists.
26. Apply the Fundamental Theorem of Line Integrals.
27. Apply Green’s theorem to line integrals around closed paths.
28. Compute the curl of a vector field and describe its implications.
29. Construct and compute surface integrals of vector fields.
30. Determine the net rate of flow of a fluid through a parameterized membrane.
31. Use Stokes’ theorem to compute surface and triple integrals.
32. Compute the divergence of a vector field and apply the divergence theorem to compute surface and triple integrals.
33. Apply techniques of vector analysis to solve problems in the sciences and engineering.
General Content
34. Write clear, correct, and complete solutions to mathematical problems utilizing proper mathematical notation and appropriate language.
35. Utilize computer algebra and graphical systems to solve problems, visualize abstract concepts, and model physical problems, as appropriate.
Degree Outcomes
Quantitative & Symbolic Reasoning: Graduates utilize mathematical, symbolic, logical, graphical, geometric, or statistical analysis for the interpretation and solution of problems in the natural world and human society.
Critical, Creative and Reflective Thinking: Graduates will evaluate, analyze, synthesize, and generate ideas; construct informed, meaningful, and justifiable conclusions; and process feelings, beliefs, biases, strengths, and weaknesses as they relate to their thinking, decisions, and creations.
Effective Communication: Graduates will be able to exchange messages in a variety of contexts using multiple methods.
Lecture Contact Hours 50
Lab Contact Hours 0
Clinical Contact Hours 0
Total Contact Hours 50
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