MATH& 264 Calculus 4 (5 credits)

1. Determine the arc length of a polar curve.

2. Determine the area enclosed by a polar curve.

3. Convert a point in three dimensional space from Cartesian coordinates to cylindrical and spherical coordinates and vice versa.

4. Identify and/or sketch a solid defined by inequalities or equations in cylindrical or spherical coordinates.

Applications of Partial Derivatives

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. Locate and test extrema using the Second Derivative Test.

8. Utilize Lagrange multipliers to optimize functions of several variables given one or more constraints.

9. Apply techniques of partial derivatives to solve problems in the sciences and engineering.

Multiple Integrals

10. Construct the double integral of a function of two variables as the limit of a Riemann sum.

11. Compute double integrals of functions of two variables over rectangular regions by identifying the relevant solid and computing its volume.

12. Compute multiple integrals over general regions whose boundaries are parameterized by curves or surfaces, utilizing the basic techniques from single variable calculus.

13. Compute double and triple integrals in polar, cylindrical, and spherical coordinates.

14. Compute the surface area of a parameterized surface via a double integral.

15. Describe and construct basic transformations and compute their Jacobians.

16. Apply the Change of Variables theorem to compute multiple integrals.

17. Apply techniques of multiple integrals to solve problems in the sciences and engineering.

Vector Calculus

18. Draw vector fields in two and three dimensions.

19. Qualitatively determine characteristics of vector fields, including periodic orbits and singularities.

20. Compute the line integral of a function and of a vector field along a parametrized curve.

21. Determine the work done by a force field in moving a mass-less particle along a parameterized trajectory.

22. Determine whether a vector field is conservative or not and construct a potential if one exists.

23. Apply the Fundamental Theorem of Line Integrals.

24. Apply Green’s theorem to line integrals around closed paths.

25. Compute the curl of a vector field and describe its implications.

26. Construct and compute surface integrals of vector fields.

27. Determine the net rate of flow of a fluid through a parameterized membrane.

28. Use Stokes’ theorem to compute surface and triple integrals.

29. Compute the divergence of a vector field and apply the divergence theorem to compute surface and triple integrals.

30. Apply techniques of vector analysis to solve problems in the sciences and engineering.

General Content

31. Write clear, correct, and complete solutions to mathematical problems utilizing proper mathematical notation and appropriate language.

32. 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

Potential Methods
A. Traditional quizzes and examinations. Computational, short answer, written proofs, and construction of counter examples.
B. Group work. In-class worksheets to explore more advanced topics from the sciences and the impact of multivariate calculus upon them. Long term projects.
C. Traditional and/or online homework.
D. Presentations – Individual and/or group presentations of problems, projects, and/or technology; oral and/or written.
E. Informal assessment – Self evaluation, peer evaluation, or teacher observation.