UMTYMP Information for Colleges and Universities
Course Equivalencies, Descriptions, and Syllabi
This page contains information for advisors or math departments at other colleges and universities who are determining credit and/or placement for graduates of the University of Minnesota Talented Youth Mathematics Program (UMTYMP).
Summary
UMTYMP courses are University of Minnesota courses covering single variable calculus, basic differential equations, linear algebra, and multivariable calculus. UMTYMP also covers topics from our department's "bridge" course, such as logic, set theory, equivalence relations, and methods of proof. At Minnesota, UMTYMP thus satisfies the lower division requirements for a math major, and students who complete UMTYMP continue directly to upper division (4000 or 5000 level) courses in the math department.
See below for specific course equivalencies, course descriptions, and syllabi. For general information about UMTYMP, see our webpage or this article from the Notices of the American Mathematical Society.
Course Equivalencies
The University of Minnesota's School of Mathematics has multiple tracks of calculus courses, including UMTYMP. If your school has already determined credit/placement for one of those sequences (e.g Math 1271, 1272, 2243, 2263), you can use the table below to determine equivalent credit/placement for UMTYMP courses. For more details about the content of each UMTYMP course, see below. If you wish to verify these course equivalences, you can contact the math department's Director of Undergraduate Studies, Prof. Bryan Mosher (mosher@umn.edu).
| Course | Calculus Track for Liberal Arts Students | Calculus Track for Science & Engineering Students | UMTYMP |
|---|---|---|---|
| Single Variable Calculus (Differentiation) | Math 1271 | Math 1371 | Math 1471 |
| Single Variable Calculus (Integration, Sequences & Series) | Math 1272 | Math 1372 | Math 1472 |
| Linear Algebra & Differential Equations | Math 2243 | Math 2373 | Math 1473 & Math 2471 |
| Multivariable Calculus | Math 2263 | Math 2374 | Math 2472 & Math 2473 |
| Bridge Course (Set theory, logic, methods of proof, precise definitions of limits) | Math 2283 | Math 2283 | Completion of UMTYMP, through Math 2473 |
Notes about Math 2283/328W: The material from our deparment's bridge course (Math 2283) is spread throughout the UMTYMP curriculum. Hence completion of the entire UMTYMP sequence (as opposed to one course) is considered equivalent to Math 2283.
There is also a writing-intensive version of the bridge course (Math 3283W). Although UMTYMP courses do not carry an official writing designation, every homework assignment includes a problem (generally a proof or other theoretical problem) which is graded for both mathematical correctness and writing. Math 2472 and Math 2473 also include large scale writing projects. By the end of the sequence, UMTYMP students have had hundreds of pages of work evaluated on the basis of writing.
Notes about Math 4990 (UMTYMP Advanced Topics): Students who finish UMTYMP but are still in high school have the option to take an Advanced Topics course. These courses typically cover upper division material; recent examples include combinatorics and computational algebraic geometry. They are not intended as replacements for full upper division courses.
Course Descriptions and Syllabi
Course syllabi, descriptions, textbooks, topics and sections covered are included here for each course. The syllabi are from the 2019-20 academic year, but apply to any UMTYMP courses in the last five years; the only changes would be dates and the specific instructors assigned to each course.
Math 1471 & 1472 (Single Variable Calculus)
Link to full syllabus: Syllabus (PDF)
Course Description. UMTYMP first-year Calculus covers the fundamentals of single-variable Calculus, including limits, differentiation, integration, and their applications. In addition, it will cover sequences and series, including Taylor and power series. While students will extensively cover computational methods, this class will incorporate deeper theoretical reasoning beyond what is typically taught in a college or AP Calculus course. Students will be expected to understand the material at this deeper conceptual level as well as being able to compute when necessary.
Math 1471: limits and limit laws; continuity; differentiability, including geometric interpretations of first and second derivatives; extreme values and optimization; differentiation rules and implicit differentiation; linear approximation; Newton's method; Mean Value Theorem and Extended Mean Value Theorem.
Math 1472: Riemann integration and techniques, including substitution and integration by parts; Trigonometric integrals and trigonometric substitution; applications, including arc length, average values, and volumes by cross sections and cylindrical shells; rigorous treatment of sequences and series.
Textbook: A custom edition of Stewart's Calculus: Concepts & Contexts book, which includes portions of the more extensive and rigorous chapters on differential equations and sequences and series from Stewart's Early Transcendentals calculus text. Sections covered:
- Stewart, Calculus CCC: Chapters 2, 3, 4, 5. Sections 6.1-6.4.
- Stewart, Calculus ET: Chapter 11
Math 1473 & Math 2471 (Foundations & Linear Algebra)
Link to full syllabus: Syllabus (PDF)
Course Description. The second year of the UMTYMP Calculus sequence begins with differential equations, including first- and second-order linear equations. We then continue on to quantifiers and precise definitions of limits of sequences and limits of functions of one variable, including rigorous proofs of limit laws. Next, we cover the basics of n-dimensional vectors and coordinates, before spending the rest of the year on a rigorous course in linear algebra.
Math 1473: differential equations; slope fields and Euler's Method; separable and linear first-order linear differential equations; second-order linear differential equations; foundations (quantifiers, logic, set theory, equivalence relations); precise definitions of limits (sequences and functions) and rigorous proofs of limit laws; 3D coordinates; vectors; dot product and cross product; parametric/vector equations of lines and planes; linear transformations.
Math 2471: Geometry and composition of linear transformations; matrix multiplication; subspaces of Rn, images, kernels, and dimension; change of bases; orthogonal transformations and orthonormal matrices; basic Gram-Schmidt process; determinants via algebra, combinatorics and geometry; eigenvalues, eigenvectors; discrete dynamical systems.
Textbooks:
- The custom calculus book described above. Sections covered: Stewart, Calculus ET: Chapters 9 (DiffEq), 12 (3D Coordinates and Vectors), and 17 (Second Order Linear DiffEq).
- Supplemental lecture notes (based on Math 3283W) on logic, set theory, and methods of proof.
- Bretchser's Linear Algebra with Applications. Sections covered: Chapters 1-4; Sections 5.1-5.3; Chapters 6-7; topics from Chapter 9 as time permits.
Math 2472 & Math 2473 (Multivariable Calculus)
Link to full syllabus: Syllabus (PDF)
Course Description. In Calculus I you learned wonderful tools for analyzing functions of a single variable. Many functions in the real world depend on more than one variable, however. In Calculus III we’ll see how the ideas from Calculus I transfer over to the world of multivariable functions. Roughly speaking the fall semester will cover differentiation, while the spring semester will cover integration. In both cases, vectors and linear transformations turn out to be vital tools. Along the way we’ll learn about parametric curves, surfaces, and solids in three (or higher) dimensions.
Math 2472: Functions of many variables; graphs and cross-sections; differential geometry of curves in 2-, 3-, and n-dimensions; parametric surfaces and tangent planes; multivariable limits; partial and directional derivatives; differentiability of multivariable functions (using the full limit definition); total derivative and linear approximations; chain rule; quadratic forms, Sylvester’s Theorem, Taylor’s Theorem, and multivariable optimization; Lagrange multipliers. Topology of Rn.
Math 2473: Multivariable integration and vector analysis; vector fields in Rn; curl and divergence; multiple integrals, including integrals of scalar valued functions and vector fields on curves and surfaces; fundamental theorem of line integrals, Green's Theorem, Stokes Theorem and the Divergence Theorem; Generalized Stokes Theorem.
Textbook: Colley's Vector Calculus, with supplemental lecture notes on topology and other topics. Sections covered: Chapters 1-7, except Section 7.4 (Maxwell's Equations), plus lecture notes on the topology of Rn.
Comparison to AP Calculus & Dual Enrollment
Advanced Placement: When comparing lists of topics, UMTYMP single variable calculus (Math 1471 & 1472) is similar to Advanced Placement BC Calculus. The courses differ in depth and rigor and, as mentioned above, all UMTYMP courses also have an additional emphasis on mathematical writing. After single variable calculus, UMTYMP covers an additional four semesters of material beyond BC Calculus.
Dual Enrollment: Many schools have specific guidelines about granting transfer credit for dual enrollment courses. UMTYMP is not part of the University of Minnesota's dual enrollment program (College in the Schools), in which high school teachers are deputized to teach University courses in their own high schools. UMTYMP courses, by contrast, are taught by University faculty, on the University campus. UMTYMP is dual enrollment only in the technical sense that students are simultaneously enrolled at the University (for UMTYMP) and their own high schools (for everything else). Students' high schools play no role in the administration, registration, or instruction of UMTYMP courses. We report UMTYMP grades to students' high schools for inclusion on their high school transcripts, so that students can document they have met mathematics requirements for high school graduation in Minnesota.
Further Questions
If you have further questions about the UMTYMP curriculum which are not answered here, please contact MathCEP (mathcep@umn.edu, 612-625-2861) or the Director of UMTYMP, Prof. Jonathan Rogness (rogness@umn.edu).
Please note: there are also high school level UMTYMP courses in Algebra, Geometry and Precalculus. Those classes do not carry University of Minnesota credit, and are not described on this page.