Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. It consists of a suite of formulas and problem solving techniques very useful for engineering and physics. Vector analysis has its origin in quaternion analysis, and was formulated by the American scientist, J. Willard Gibbs [1]. It concerns vector fields, which associate a vector to every point in space, and scalar fields, which associate a scalar to every point in space. For example, the temperature of a swimming pool is a scalar field: to each point we associate a scalar value of temperature. The water flow in the same pool is a vector field: to each point we associate a velocity vector.
calculus chain change continuous curl derivative differential dimension function geometry mathematics rule variables vector

Note: The summary of gradient, curl and divergence is particularly good.
• College Algebra - Math 116 - Lecture Notes by James Jones
These notes were written during the Fall 1997 semester to accompany Larson's College Algebra: A Graphing Approach, 2nd edition text. We have moved on to Larson's 4th edition and some sections have changed but I have left them where they are since many people on the Internet find these useful resources. The notes were updated in the Fall 2003 semester to use Cascading Style Sheets and validate as XHTML 1.0 strict web pages. If your browser doesn't support CSS, certain pages (especially those with matrices) will not display properly.

Note: A nice book on basic algebra operations.
• Differential Equations - Wikibooks, collection of open-content textbooks
This book aims to lead the reader through the topic of differential equations, a vital area of modern mathematics and science. It is hoped that this book will provide information about the whole area of differential equations, but for the moment it will concentrate on the simpler equations.
applications bernoulli deqs differential equations exact existence first formation frobenius linear order second solution the

Note: This stuff I found to be rather good: http://en.wikibooks.org/wiki/Differential_Equations/First_Order
• Elementary Topology: Second Edition by Michael C. Gemignani
Superb introduction to rapidly expanding area of mathematical thought. Fundamentals of metric spaces, topologies, convergence, compactness, connectedness, homotopy theory and other essentials. Numerous exercises, plus section on paracompactness and complete regularity. References throughout. Includes 107 illustrations.
Amazon.com: Elementary Topology: Second Edition (Dover Books on Mathematics) (9780486665221): Michael C. Gemignani: Books
0486665224 20724441 books dover edition elementary gemignani geometry mathematics michael publications science second topology

Note: Recommended by John Banks.
• The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics - Marcus du Sautoy
The quest to bring advanced math to the masses continues with this engaging but quixotic treatise. The mystery in question is the Riemann Hypothesis, named for the hypochondriac German mathematician Bernard Reimann (1826-66), which ties together imaginary numbers, sine waves and prime numbers in a way that the world's greatest mathematicians have spent 144 years trying to prove. Oxford mathematician and BBC commentator du Sautoy does his best to explain the problem, but stumbles over the fact that the Riemann Hypothesis and its corollaries are just too hard for non-tenured readers to understand. He falls back on the staples of math popularizations by shifting the discussion to easier math concepts.
Amazon.com: The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (9780060935580): Marcus du Sautoy: Books
0060935588 acamp_book_new_00609355 greatest harper marcus mathematics music mystery perennial primes sautoy searching solve the

Note: Availble at Melb uni maths library: http://cat.lib.unimelb.edu.au/search/X?SEARCH=music+of+the+primes&searchscope=30&SORT=D&searchType=X
• Topology (2nd Edition) - by James Munkres - amazon.com
This introduction to topology provides separate, in-depth coverage of both general topology and algebraic topology. Includes many examples and figures. GENERAL TOPOLOGY. Set Theory and Logic. Topological Spaces and Continuous Functions. Connectedness and Compactness. Countability and Separation Axioms. The Tychonoff Theorem. Metrization Theorems and paracompactness. Complete Metric Spaces and Function Spaces. Baire Spaces and Dimension Theory. ALGEBRAIC TOPOLOGY. The Fundamental Group. Separation Theorems. The Seifert-van Kampen Theorem. Classification of Surfaces. Classification of Covering Spaces. Applications to Group Theory. For anyone needing a basic, thorough, introduction to general and algebraic topology and its applications.

Note: John Banks recommended it. More formal than gemignani's 'elementary topology'.
• Elements of general topology by Donald Bushaw (2nd edition)
Author: Bushaw, Donald. Title: Elements of general topology. Published: New York : J. Wiley, [1963].

Note: A really good, well worded account on point set topology, gives a good sketch of the motivation behind definitions. LTU Bundy's got it: https://alpha2.latrobe.edu.au/patroninfo/1119178/item&1287918
• Proof: "any topological space with the fixed point property is connected" - PlanetMath
Theorem Any topological space with the fixed-point property is connected. Proof. We will prove the contrapositive. ....

Note: A decent explanation of the fixed point property: http://planetmath.org/encyclopedia/FixedPointProperty.html
• The Math Forum @ Drexel University
The Math Forum Is... ... the leading online resource for improving math learning, teaching, and communication since 1992. _We are teachers, mathematicians, researchers, students, and parents using the power of the Web to learn math and improve math education. _We offer a wealth of problems and puzzles; online mentoring; research; team problem solving; collaborations; and professional development. Students have fun and learn a lot. Educators share ideas and acquire new skills. _
The Math Forum is the comprehensive resource for math education on the Internet. Some features include a K-12 math expert help service, an extensive database of math sites, online resources for teaching and learning math, plus much more.
algebra area calculus curriculum education geometry help homework math mathematics perimeter trig trigonometry volume
by 6 users
• A Beautiful Mind's John Nash is less complex than the real one. - By Chris Suellentrop - Slate Magazine
Here's what's true in Ron Howard's movie A Beautiful Mind—or, at least, here's what corresponds to Sylvia Nasar's biography of the same name: The mathematician John Forbes Nash Jr. attended graduate school at Princeton, where he was arrogant, childish, and brilliant. His doctoral thesis on the so-called "Nash equilibrium" revolutionized economics. Over time, he began to suffer delusions. He was hospitalized for paranoid schizophrenia, administered insulin shock therapy, and released...

• Bijection - wikipedia
In mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property that, for every y in Y, there is exactly one x in X such that f(x) = y. Alternatively, f is bijective if it is a one-to-one correspondence between those sets; i.e., both one-to-one (injective) and onto (surjective).[1] (See also Bijection, injection and surjection.)
2009 articles axiomatic bijection bijective cardinal cardinality from lacking march number numeration proof set sources theory
• Bijection, injection and surjection - Wikipedia
In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.
and axiom bijection bijective bourbaki cardinality category choice codomain domain function injection mathematics surjection

Note: Need to learn this.
• Fixed point property - Wikipedia
In mathematics, a topological space X has the fixed point property if all continuous mappings from X to X have a fixed point.
1932 brouwer category closed compact concrete continuous disc fixed interval mathematics point property space theorem

Note: In mathematics, a fixed point (sometimes shortened to fixpoint) of a function is a point that is mapped to itself by the function... http://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29
• Fixed-point theorem - Wikipedia
In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most generally useful in mathematics. The Banach fixed point theorem gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).
algebraic atiyahbott banach borel bourbaki-witt brouwer caristi church-turing fixed fixed-point point space th theorem topology
• Introduction to Mathematical Logic by Elliott Mendelson
I was sufficiently fortunate to have taken Professor Emeritus Mendelson's famous logic course at Queens College, the City University of New York, just two semesters before his retirement. I was, and continue to be, astonished by Dr. Mendelson's precise yet easy style, and the beautifully efficient organization of the subjects. Everything from the expository prose to the system of notational conventions has been carefully thought through so as to make the book both very substantive and very readable. In my opinion, it's the best introduction to serious mathematical logic currently on the market, and thanks to the genius of its author, it is likely to remain so for a long time. The buyer will not be disappointed.
Amazon.com: Introduction to Mathematical Logic, Fourth Edition (9780412808302): E. Mendelson: Books
0412808307 7063817 and edition fourth introduction logic mathematical mathematics mendelson problems science springer symbolic

Note: LTU lib has it (2nd edition): http://library.latrobe.edu.au/search/XIntroduction+to+Mathematical+Logic+Elliott+Mendelson&searchscope=1&SORT=A/XIntroduction+to+Mathematical+Logic+Elliott+Mendelson&searchscope=1&SORT=A&extended=0&SUBKEY=Introduction%20to%20Mathematical%20Logic%20Elliott%20Mendelson/1% ...more2C3%2C3%2CB/frameset&FF=XIntroduction+to+Mathematical+Logic+Elliott+Mendelson&SORT=A&1%2C1%2C
• Musings of the Masters: An Anthology of Miscellaneous Reflections by Raymond Ayoub
The anthology is a collection of articles contiguous to the humanities written by renowned mathematicians of the twentieth century. The articles cover a variety of topics that, for want of a better name, shall be referred to as humanistic. An important criterion, thereby limiting the choice, is that the articles should be accessible to the literate reader who may or may not have technical knowledge of mathematics. The articles span roughly a century in time and a wide range in subject. They are by mathematicians acknowledged by their peers as outstanding creators whose work has added richly to the discipline. Each article is preceded by a brief biographical sketch of the author and a brief indication of the content.
Amazon.com: Musings of the Masters: An Anthology of Mathematical Reflections (Spectrum) (9780883855492): Raymond Ayoub: Books
0883855496 amer anthology assn ayoub bkk-04214825-m masters mathematical musings raymond reflections so sociology spectrum the

Note: Melbourne maths library got it (of course!): http://cat.lib.unimelb.edu.au/search/XMusings+of+the+Masters+An+Anthology+of+Miscellaneous+Reflection&f=&searchscope=30&m=&l=&Da=&Db=&p=&SORT=D/XMusings+of+the+Masters+An+Anthology+of+Miscellaneous+Reflection&f=&searchscope=30&m=&l=&Da=&Db=&p=&SORT=D&SUBK ...moreEY=Musings%20of%20the%20Masters%20An%20Anthology%20of%20Miscellaneous%20Reflection/1%2C32000%2C32000%2CB/frameset&FF=XMusings+of+the+Masters+An+Anthology+of+Miscellaneous+Reflection&SORT=D&1%2C1%2C
• Open problems in topology edited by Jan van Mill, George M. Reed.
Title: Open problems in topology / edited by Jan van Mill, George M. Reed. Published: Amsterdam [Netherlands] ; New York : North-Holland ; New York, N.Y., U.S.A : Distributors for the U.S. and Canada, Elsevier Science Pub. Co., 1990.

Note: Doesn't seem like they got to their third edition, despite saying it was due around 1995. Need to know a lot more topology before I can understand the problems.
• Range (mathematics) - Wikipedia
In mathematics, the range of a function is the set of all "output" values produced by that function. Sometimes it is called the image, or more precisely, the image of the domain of the function....The range should not be confused with the codomain B. The range is a subset of the codomain, but is not necessarily equal to the codomain, since there may be elements of the codomain which are not elements of the range. The codomain is sometimes taken to be the range, but more often is some standard set, such as the real numbers or the complex numbers, which contains the range. A function whose range equals its codomain is called onto or surjective.
and bijection cartesian codomain coordinate domain exponentiation function image injection mathematics range surjection system
• Separation axiom - Wikipedia
In topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms. These are sometimes called Tychonoff separation axioms, after Andrey Tychonoff. The separation axioms are axioms only in the sense that, when defining the notion of topological space, you could add these conditions as extra axioms to get a more restricted notion of what a topological space is. The modern approach is to fix once and for all the axiomatization of topological space and then speak of kinds of topological spaces.

Note: See also: The precise meanings of the terms associated with the separation axioms has varied over time, as explained in History of the separation axioms... http://en.wikipedia.org/wiki/History_of_the_separation_axioms