Introductory University Mathematics Volume 1 Pdf

[Malene Mederios]

Math 18 - Foundations for Calculus (2 units, S/NC, Fall only) covers the mathematical background and fundamental skills necessary for success in calculus and other college-level quantitative work. Topics include ratios, unit conversions, functions and graphs, polynomials and rational functions, exponential and logarithm, trigonometry and the unit circle, and word problems. Class sessions are a mix of lecture and worksheets.

This series covers differential calculus, integral calculus, and power series in one variable. It can be started at any point in the sequence for those with sufficient background. See the detailed list of topics for the Math 20 series.

Covers properties and applications of limits, continuous functions, and derivatives. Calculations involve trigonometric functions, exponentials, and logarithms, and applications include max/min problems and curve-sketching.

Covers properties and applications of integration, including the Fundamental Theorem of Calculus and computations of volumes, areas, and arc length of parametric curves. An introduction to some basic notions related to differential equations (such as exponential growth/decay and separable equations) is also given.

Covers limits at infinity and unbounded functions in the context of integration as well as infinite sums, including convergence/divergence tests and power series. Taylor series and applications are also covered.

The content of Math 21 (improper integrals, infinite series, and power series) is essentially the material of BC-level AP calculus not in the syllabus of AB-level AP calculus nor in IB Higher Level math. The math placement diagnostic results do not waive Math 21 requirements, since the diagnostic has no exam security; its feedback is purely advisory. Knowledge of Math 21 content is fundamental to university-level quantitative work, and is expected by the outside world for anyone earning a degree in a quantitative field here. This is an enforced requirement to enroll in Math 51 or CME 100; for more details, click the button above.

Math 51- Linear Algebra, Multivariable Calculus, and Modern Applications (5 units) covers linear algebra and multivariable differential calculus in a unified manner alongside applications related to many quantitative fields. This material includes the basic geometry and algebra of vectors, matrices, and linear transformations, as well as optimization techniques in any number of variables (involving partial derivatives and Lagrange multipliers).

The unified treatment of both linear algebra (beyond dimension 3 and including eigenvalues) and multivariable optimization is not covered in a single course accessible to non-majors anywhere else. Many students who learn some multivariable calculus before arriving at Stanford find Math 51 to be instructive to take due to its broad scope and synthesis of concepts. If you want transfer credit to substitute for Math 51 then you will likely need two courses (one on multivariable calculus, one on linear algebra).

Math 53- Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications (5 units) develops core concepts, examples, and results for ordinary differential equations, and covers important partial differential equations and Fourier techniques for solving them. This uses both linear algebra and matrix derivative material from Math 51.

This series provides the necessary mathematical background for majors in all disciplines, especially for the Natural Sciences, Mathematics, Mathematical and Computational Science, Economics, and Engineering.

Math 61CM-62CM-63CM- Modern Mathematics: Continuous Methods (5 units each) This proof-oriented three-quarter sequence covers the material of 51, 52, 53, and additional advanced calculus, higher-dimensional geometry, and ordinary and partial differential equations. This provides a unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know single-variable calculus very well and have an interest in a theoretical approach to the subject.

This proof-oriented three-quarter sequence covers the same linear algebra and multivariable optimization material as the 60CM-series but draws its motivation from topics in discrete math rather than from the more analytic topics as in the 60CM-series. Its discrete math coverage includes combinatorics, probability, some basic group theory, number theory, and graph theory. Students should have an interest in a theoretical approach to the subject.

Many 100-level mathematics courses assume familiarity with writing proofs, and if you plan to be a Math major then you should learn proof writing as soon as possible. Here is a list of courses to begin learning proof-writing:

Math 104 also provides an introduction to proof-writing, but not at the same level as the above courses (a variety of proofs are covered, but students are not expected to write proofs of their own at the same level as some of those shown in class).

For more information about these courses see ExploreCourses for course descriptions and schedule. If you have further questions about which course to take, contact your academic advisor, or our Director of Undergraduate Studies.

There was a time when I avoided math proofs, but now I am starting to enjoy them. I am taking Intro to Linear Algebra and am falling in love with proofs. Are there any introduction to mathematical proofs books that blow the others out of the water?

George Polya's How to Solve It immediately comes to mind. I know many now fantastic pre-mathematicians who learned calculus and the basics of analysis from Spivak's Calculus and even if you know the material to go back and do it again in a formal way is very healthy. In addition Proofs from THE BOOK was mentioned above and was recommended to me by Ngo Bao Chao when I asked about books to study problem-solving techniques from. I don't mean to come off as name-dropping but I feel that (as he is a fields medalist) his advice is worth heeding. I, personally, really liked it.

However I have to make note that I think if you'd phrased your question as "should I read a book about proofs to learn proofs" my response would be an emphatic no. In my experience if you don't see proofs by doing some fun mathematics you will not get much better about doing them yourself. Just reading about how to prove things can only get you so far before you're sort of stumped as to how to proceed. I would say the better approach is to find a rigorous treatment of a subject that you're very interested in, and read that, following along with the proofs of the theorems in the book and eventually trying to do them yourself without looking at the proofs given.

One option is to read an introductory book on a topic that interests you. For example, if you are interested in number theory, you can read Harold Stark's An Introduction to Number Theory. Depending on your motivation and degree of comfort reading proofs at this level, something like this might be a good option - an "introduction to proofs" book isn't a necessity for everyone.

However, if you want a book that is geared specifically for those who are just starting out with rigorous math and are still getting used to proofs, you might enjoy Journey into Mathematics: An Introduction to Proofs by Joseph Rotman. Unlike some such books, it doesn't dwell on trivialities about logic and sets. Instead, it discusses interesting yet accessible topics in elementary mathematics like Pythagorean triples, the number $\pi$, and cubic and quartic equations. Along the way, it introduces important concepts such as proof by induction, the formal definition of convergence of a sequence, and complex numbers. The book makes use of calculus, taking advantage of the fact that most North American students at this "transition to advanced mathematics" stage have already had courses in calculus. But what you will remember after reading it ought to be the actual mathematics in it, so you hopefully won't feel as if you've wasted your time.

It's not that sets and logic can't be interesting in themselves, but usually the more interesting aspects of these subjects can only be appreciated once a learner is well acquainted with mathematical methods in general.

A book used at my university in a first-year intro to mathematical thinking course is Tamara J. Lakin's The Tools of Mathematical Reasoning. It covers introductory logic, proofs, sets, functions, number theory, relations, finite and infinite sets, and the foundations of analysis. I liked it.

Introduction to limits and derivatives as preparation for further courses in calculus. Fundamental functions (polynomials, rational functions, exponential, logarithmic, trigonometric) and their graphs will be also reviewed. Other topics include tangent and normal lines, linearization, computing area and rates of change. The emphasis will be on learning to think independently and creatively in the mathematical setting.

One semester survey of the major concepts and computational techniques of calculus including limits, derivatives and integrals. Emphasis on basic examples and applications of calculus including approximation, differential equations, rates of change and error estimation for students who will take no further calculus. Prerequisites: MAT100 or equivalent. Restrictions: Cannot receive course credit for both MAT103 and MAT102. Provides adequate preparation for MAT175. Three classes.

First semester of calculus. Topics include limits, continuity, the derivative, basic differentiation formulas and applications (curve-sketching, optimization, related rates), definite and indefinite integrals, the fundamental theorem of calculus. Prerequisite: MAT100 or equivalent.

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