MATH 3334 - Advanced Multivariable Calculus - University of Houston

# MATH 3334 - Advanced Multivariable Calculus

***This is a course guideline.  Students should contact instructor for the updated information on current course syllabus, textbooks, and course content***

Prerequisites: MATH 3333

Course Description: Basic theory underlying multivariable calculus: a brief overview of the topology of n-space, limits, continuity and differentiation of functions of several variables, Taylor’s theorem, the inverse and implicit function theorems, integration

Recommended Text: “Advanced Calculus”, Edition: 3; R. Creighton Buck. Waveland Press, Inc. ISBN: 9781577663027

This is a first course in the basic theory and analysis underlying n-dimensional calculus. It will be important in preparing the student for any later courses involving analysis or calculus. There will be an emphasis on definitions, theorems and proofs.

Suggested Syllabus

I. Topology of Rn

Including vectors in Rn, the Cauchy-Schwarz and triangle inequalities, convergent sequences in Rn, open and closed sets, the closure and boundary, (a short discussion of) compactness, connectedness, limits and continuity, continuity with respect to the properties of compactness and connectedness and the Extreme Value Theorem. (The instructor may have to supplement the discussion of the topology of Rn)

II. Multivariable differentiation.

Including scalar functions of class C1 and Ck on an open set in Rn, the equality of mixed partial derivatives, Taylor's Theorem, extrema and the first and second derivative tests, the matrix-vector definition of the derivative of a function f maps D subset of Rn to Rn (2.3 and 2.7 in the recommended text) and its implications, the Chain Rule and at least a statement and an illustration of the Inverse and Implicit Function Theorems (which are fully proved in Math 4332 ) .

III. Multivariable integration.

Including Riemann sums, the double and triple integral and their basic properties, Fubini's theorem, the Jacobian determinant and the change of variables formula. The emphasis is on rigorous definitions and (selected) proofs. Generalizations of the Fundamental Theorem of the Calculus, such as the Fundamental Theorem of Line Integrals, or the Divergence Theorem may be discussed if time permits.

CSD Accommodations:

Accommodation Forms: Students seeking academic adjustments/auxiliary aids must, in a timely manner (usually at the beginning of the semester), provide their instructor with a current Student Accommodation Form (SAF) (paper copy or online version, as appropriate) from the CSD office before an approved accommodation can be implemented.

Details of this policy, and the corresponding responsibilities of the student are outlined in The Student Academic Adjustments/Auxiliary Aids Policy (01.D.09) document under [STEP 4: Student Submission (5.4.1 & 5.4.2), Page 6]. For more information please visit the Center for Students with Disabilities Student Resources page.

Additionally, if a student is requesting a (CSD approved) testing accommodation, then the student will also complete a Request for Individualized Testing Accommodations (RITA) paper form to arrange for tests to be administered at the CSD office. CSD suggests that the student meet with their instructor during office hours and/or make an appointment to complete the RITA form to ensure confidentiality.

*Note: RITA forms must be completed at least 48 hours in advance of the original test date. Please consult your counselor ahead of time to ensure that your tests are scheduled in a timely manner. Please keep in mind that if you run over the agreed upon time limit for your exam, you will be penalized in proportion to the amount of extra time taken.

UH CAPS

Counseling and Psychological Services (CAPS) can help students who are having difficulties managing stress, adjusting to college, or feeling sad and hopeless. You can reach (CAPS) by calling 713-743-5454 during and after business hours for routine appointments or if you or someone you know is in crisis. No appointment is necessary for the "Let's Talk" program, a drop-in consultation service at convenient locations and hours around campus.