These are semi-official #math book recommendations for various topics. These are all personal recommendations of channel regulars. This means that given the collective experience of the channel, these are the books to read. We made an arbitrary split between the mathematics before and after calculus. All of the former will be found in Pre-College Mathematics section. All of the latter will be found each under its own subject listing. We also have a section devoted to books of general interest.
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These books are intended for a general audience. history/philosophy of math, 'how to solve it'
The History of Calculus and Its Conceptual Development (http://www.amazon.com/gp/product/0486605094/) by Boyer
A History of Mathematics (http://www.amazon.com/gp/product/0471543977/) by Boyer
A Mathematician's Apology (http://www.amazon.com/gp/product/0521427061/) by Hardy
This category is the catch-all for topics generally preceding calculus. This includes high school algebra, geometry and trigonometry. Unfortunately decent books are hard to find. The recommendations here are not as personal and less authoritative than the others. They are made to be helpful, but should be taken with a grain of salt.
Schaum's outlines -- I used various ones for college-level math and they were useful, and the calculus one below is well-recommended. But some are good; some are bad.
- Algebra the Easy Way (http://www.amazon.com/gp/product/0764119729/) by Downing
- Geometry the Easy Way (http://www.amazon.com/gp/product/0764101102/) by Leff
- Trigonometry the Easy Way (http://www.amazon.com/gp/product/0764113607/) by Downing
- Precalculus the Easy Way (http://www.amazon.com/gp/product/0764128922/) by Leff
The book Calculus the Easy Way (http://www.amazon.com/gp/product/0812091418/) by Downing is an easy and intuitive introduction to calculus. It is on this basis (and only this basis) that I recommend the other books in the series.
In course titles, often called "abstract" or "modern" algebra to distinguish it from high school algebra, but known simply as algebra otherwise, this subject analyzes groups, rings, and fields. These are all algebraic structures but with sometimes subtly different properties. See also: commutative algebra, linear algebra, representation theory.
Abstract Algebra (http://www.amazon.com/gp/product/0471368571/) by Dummit and Foote
Groups and Representations (http://www.amazon.com/gp/product/0387945261/) by Alperin and Bell
Field and Galois Theory (http://www.amazon.com/gp/product/0387947531/) by Morandi
Topics in Algebra (http://www.amazon.com/Topics-Algebra-I-N-Herstein/dp/0471010901/ref=sr_1_1?ie=UTF8&s=books&qid=1272768920&sr=1-1) by Herstein
A quote from Gallian's book states "A whole generation of textbooks and an entire generation of mathematicians, myself included, have been profoundly influenced by that text" - Georgia Benkart. Also see Abstract Algebra by the same author for a lower level treatment.
A first Course in Abstract Algebra (http://www.amazon.com/First-Course-Abstract-Algebra-7th/dp/0201763907/ref=sr_1_1?ie=UTF8&s=books&qid=1272769054&sr=1-1) by Fraleigh
A good introduction to abstract algebra. More gentle than Herstein.
Contemporary Abstract Algebra (http://www.amazon.com/Contemporary-Abstract-Algebra-Joseph-Gallian/dp/0547165099/ref=sr_1_2?ie=UTF8&s=books&qid=1272769054&sr=1-2) by Gallian
Another introduction to abstract algebra. Similar to Fraleigh, but with some topic reorderings. Many problems and hints in back of text. Many biographies of mathematicians, some GAP usage.
A first Course in Abstract Algebra with Applications (http://www.amazon.com/First-Course-Abstract-Algebra-3rd/dp/0131862677/ref=sr_1_1?ie=UTF8&s=books&qid=1272769173&sr=1-1) by Rotman
A introductory abstract algebra text. The ordering is more awkward than the prior texts, but sections are fairly well written.
Algebra (http://www.amazon.com/Algebra-Michael-Artin/dp/0130047635/ref=sr_1_1?ie=UTF8&s=books&qid=1272769272&sr=1-1) by Artin
Classically, algebraic geometry was the study of polynomials, and more importantly the geometry of their solution sets. In modern times, while this statement remains true, the tremendous technical power of the methods of modern algebraic geometry (ex: sheaves, cohomology, stacks, Grothendieck topologies, etc) has ensured that the influence of algebraic geometry on other subjects in mathematics (notably number theory, algebraic topology and representation theory) is impossible to overestimate.
A few introductory books on algebraic geometry are:
Algebraic Geometry: A First Course (http://www.amazon.com/gp/product/0387977163/) by Harris
Undergraduate Algebraic Geometry (http://www.amazon.com/gp/product/0521356628/) by Reid
A basic treatment of scheme theory and (Zariski) sheaf cohomology can be found in:
EGA (http://cohomology.princeton.edu/books/Math/EGA_SGA/EGA/) by Grothendieck
Algebraic Geometry (http://www.amazon.com/gp/product/0387902449) by Hartshorne
Algebraic Geometry and Arithmetic Curves (http://www.amazon.com/gp/product/0198502842) by Liu
Many important results of Grothendieck, besides those in EGA, can be found in SGA (http://cohomology.princeton.edu/books/Math/EGA_SGA/SGA/).
Mumford's Abelian Varieties (http://www.amazon.com/gp/product/0195605284) is perhaps the most beautiful text on varieties which are the algebraic analogue of the Lie groups of differential geometry. It does, however, lack arithmetic applications as it always works over algebraically closed based fields.
Champs Algebrique (http://www.amazon.com/gp/product/3540657614/) by Laumon and Morret-Baily (LMB) is the canonical reference when it comes to the theory of algebraic stacks (which originates in the seminal paper (http://www.numdam.org/numdam-bin/item?id=PMIHES_1969__36__75_0) of Deligne and Mumford, two Fields medallists). However, if you're not a Fields medallist, the french book is an incredibly hard read. Fortunately, many people who wish to learn about stacks are not Fields medallists and are extremely good expositors. So there are plenty of places on the web where you can find a basic readable introduction to the theory of stacks, such as this (http://arxiv.org/abs/math/0412512) one. One must keep in mind, however, that these are only introductions and that LMB still remains the standard reference. If, on the other hand, you find that LMB is not general enough or if you're more categorically inclined, Giraud's book (``Cohomologie non-abelienne"), which talks in depth about fibered categories and stacks (but not *algebraic* stacks), might be worth a look.
Claire Voisin's new books (1 (http://www.amazon.com/gp/product/052180260) and 2 (http://www.amazon.com/gp/product/0521802830)) beautifully exhibit the applications of algebraic geometry to differential and complex geometry. It is only a matter of time before these books take their place besides Griffiths' and Harris' beautiful text (http://www.amazon.com/gp/product/0471050598/) as the standard references for complex algebraic geometry.
For those interested in arithmetic applications, there's no shortage of good books either. Silverman's twins (1 (http://www.amazon.com/gp/product/0387962034) and 2 (http://www.amazon.com/gp/product/0387943285)) are wonderfully readable introductions to the arithmetic theory of elliptic curves, especially since they avoid the high-brow methods of the theory of abelian varieties when possible, thereby making the subject more accessible. The proceedings (http://www.amazon.com/gp/product/0387963111) of the conference that followed Faltings' stunning proof of the Mordell conjecture are full of beautiful surveys on various topics of interest to arithmetic geometers, even those who don't care about Faltings' proof. The Neron models book (http://www.amazon.com/gp/product/0387505873) too is full of expositions on various topics of interest to arithmetic geometers.
Grothendieck dreamt of a world of motives in the 60s and, almost half a century later, we're still trying to realise this dream. An extremely nice summary of the progress in this direction, up until the mid 90's, can be found in the twin-volume Seattle conference proceedings (http://www.ams.org/bookstore-getitem/item=pspum-55.s). The articles in these proceedings are, generally, very well-written and, as such, provide concise introductions to various subjects and, consequently, might be helpful to even those with no direct interest in motives. In the subsequent years, Voevodsky has come up (http://www.amazon.com/Cycles-Transfers-Motivic-Homology-Theories/dp/0691048150/sr=8-1/qid=1161626911/ref=sr_1_1/002-2929973-5413605?ie=UTF8&s=books) with a conjectural motivic cohomology theory which has already found spectacular applications (ex: the Milnor conjecture)
This (http://www.amazon.com/gp/product/082183424X) recently published (bilingual) book contains numerous letters that Serre and Grothendieck, two of the founding fathers of modern algebraic geometry, exchanged in the earlier part of the second half of the twentieth century. For a student of modern algebraic geometry, reading these letters can be a highly rewarding experience on many counts. Firstly, they expose the mistakes these awe-inspiringly smart mathematicians made, thereby making them more human. Secondly, they beautifully elucidate how many of the great ideas in algebraic geometry (ex: motives) originated. Lastly, they also help eliminate the misconeption that mathematicians do all their work alone (stuck up in attic, sometimes!); indeed, these letters are one of the best examples of the kind of collaboration that takes place at the highest level of the subject. All in all, a wonderful read for anyone with more than a passing interest in algebraic geometry.
Algebraic topology systematically uses algebra to solve topological problems.
Algebraic Topology (http://www.amazon.com/gp/product/0805335579/) by Greenberg and Harper
Algebraic Topology (http://www.math.cornell.edu/~hatcher/#ATI) by Allen Hatcher - Freely Available online, leaves a lot to problems
A basic course in algebraic topology (http://www.amazon.com/basic-course-algebraic-topology-127/dp/038797430X/ref=sr_1_7?s=books&ie=UTF8&qid=1333586048&sr=1-7) by William Massey (combines and updates most of the material from Algebraic Topology: An Introduction and Singular Homology Theory by the same author)
Algebraic Topology (http://www.amazon.com/Algebraic-Topology-Edwin-H-Spanier/dp/0387944265/ref=sr_1_11?s=books&ie=UTF8&qid=1333586048&sr=1-11) by Edward Spanier
Elements of Algebraic Topology (http://www.amazon.com/Elements-Algebraic-Topology-James-Munkres/dp/0201627280/ref=sr_1_12?s=books&ie=UTF8&qid=1333586048&sr=1-12) by Munkres (See also the second part of Topology 2e, by the same author)
Calculus classes cover a wide range of topics from differentiation and integration of single variable functions to differentiation and integration of functions of many variables. See also the real analysis books.
Calculus (http://www.amazon.com/gp/product/0914098896/) by Spivak
Calculus: One and Several Variables (http://www.amazon.com/gp/product/0471231207/) by Salas, Hille, and Etgen
Schaum's Outline of Advanced Calculus (http://www.amazon.com/gp/product/0071375678/) by Wrede and Spiegel
Notes (http://tutorial.math.lamar.edu/Classes/CalcI/CalcI.aspx) by Paul Dawkins (This site also has notes for basic linear algebra, differential equations and basic algebra)
Undergraduate Commutative Algebra (http://www.amazon.com/gp/product/0521458897/) by Reid
Introduction to Commutative Algebra (http://www.amazon.com/gp/product/0201407515/) by Atiyah and Macdonald
Commutative Ring Theory (http://www.amazon.com/gp/product/0521367646/) by Matsumura
Complex analysis is the study of differentiable functions of a complex variable. This differentiability is much stronger than the differentiability of a function of a real variable, and the theory is much richer as a result.
Complex Analysis in One Variable (http://www.amazon.com/gp/product/0817641645/) by Narasimhan
Complex Variables and Applications (http://www.amazon.com/gp/product/0072872527/) by Brown and Churchill
An excellent text for students of physical sciences & engineering.
Topics in Complex Analysis (http://www.amazon.com/gp/product/038794754X/) by Andersson
Basic Complex Analysis (http://www.amazon.com/Basic-Complex-Analysis-Jerrold-Marsden/dp/071672877X/ref=sr_1_1?ie=UTF8&s=books&qid=1272769338&sr=1-1) by Marsden & Hoffman
Alternative to Brown & Churchill.
Differential Equations Problem Solver (http://www.amazon.com/gp/product/0878915133/) by REA
Boyce & DiPrima (older editions better)
The Analysis of Linear Partial Differential Operators I-IV (http://www.amazon.com/Analysis-Linear-Partial-Differential-Operators/dp/3540006621/ref=sr_1_2/105-6622941-2245208)
Up to date collection on modern theory of linear PDE (ie based on distributions). Should be considered very theoretic and is certainly the wrong place to start, but a good reference. The text is well written but terse.
Differential Equations With Boundary-Value Problems (http://www.amazon.com/Differential-Equations-Boundary-Value-Problems-Tutorial/dp/0534418872/ref=ed_oe_h) by Zill & Cullen
A good text for engineering undergraduates. Older editions are preferable (and much cheaper!)
Applied Partial Differential Equations (http://www.amazon.com/Applied-Partial-Differential-Equations-4th/dp/0130652431/ref=sr_1_1?ie=UTF8&s=books&qid=1272769669&sr=1-1) by Haberman
Good introduction to undergraduate applied partial differential equations. Suitable for applied math, physical sciences and engineering.
Partial Differential Equations (http://www.amazon.com/Partial-Differential-Equations-Walter-Strauss/dp/0470054565/ref=sr_1_1?ie=UTF8&s=books&qid=1272769715&sr=1-1) by Strauss
An alternative to Haberman. Similar in level & material.
Beginning Partial Differential Equations (http://www.amazon.com/Beginning-Partial-Differential-Equations-Mathematics/dp/0470133902/ref=sr_1_1?ie=UTF8&s=books&qid=1272769762&sr=1-1) By O'Neil
Lower level than Haberman. Emphasizes more method of characteristics in beginning.
Partial Differential Equations of Applied Mathematics (http://www.amazon.com/Partial-Differential-Equations-Applied-Mathematics/dp/0471690732/ref=sr_1_1?ie=UTF8&s=books&qid=1272769872&sr=1-1) by Zauderer
A higher level text than Haberman.
Differential topology is the natural generalization of calculus of many variables. Calculus being essentially local, the object of study in differential topology — known as a manifold — locally acts and feels like ordinary Euclidean space. As such, one can study manifolds and functions on them in a manner very similar to the way one does in multivariable calculus. One can speak of derivatives and integrals, use conditions on the derivative to determine extrema, and much more... That being said, there are global properties of manifolds that are beyond what ordinary calculus can see.
Topology from the Differentiable Viewpoint (http://www.amazon.com/gp/product/0691048339/) by Milnor
This short book by Milnor covers the basics: manifolds, smooth maps and their derivatives, Sard's theorem, vector fields and degree, and Hopf's index theorem. Milnor is one of the best expositors in the world of mathematics, and this book is a prime example.
Differential Topology (http://www.amazon.com/gp/product/0132126052/) by Guillemin and Pollack
Calculus on Manifolds (http://www.amazon.com/gp/product/0805390219/) by Spivak
Essential Results of Functional Analysis (http://www.amazon.com/gp/product/0226983382/) by Zimmer
Elements of Information Theory (http://www.amazon.com/gp/product/0471062596/sr=1-1/qid=1139005216/ref=pd_bbs_1/002-0300919-4226435?%5Fencoding=UTF8) by Cover and Thomas
This is a common text for approximately graduate level electrical engineering departments and should be a first resource.
Information Theory, Inference, and Learning Algorithms (http://www.inference.phy.cam.ac.uk/mackay/itila/) by David MacKay ( freely available from author for computer viewing )
This text concentrates more on coding than the prior, and does a wide variety of applications. It is less engineering oriented (no network information theory, rate distortion theory, etc.)
Information Theory and Reliable Communication (http://www.amazon.com/Information-Theory-Reliable-Communication-Gallager/dp/0471290483) by Gallager
A classic in the field for almost 45 years.
Information Theory (http://www.amazon.com/Introduction-Information-Theory-John-Pierce/dp/0486240614/ref=sr_1_1?ie=UTF8&s=books&qid=1272768241&sr=1-1) by Pierce
Rather non-mathematical in nature. Broad overview, Dover book.
Information Theory (http://www.amazon.com/Information-Theory-Robert-B-Ash/dp/0486665216/ref=sr_1_4?ie=UTF8&s=books&qid=1272768241&sr=1-4) by Ash
A more mathematically oriented dover book. Has good problems and solutions. Fairly self contained with good appendices.
The Mathematical Theory of Communication (http://www.amazon.com/Mathematical-Theory-Communication-Claude-Shannon/dp/0252725468/ref=sr_1_8?ie=UTF8&s=books&qid=1272768241&sr=1-8) by Shannon
Shannon pioneered information theory. Available freely online.
Information Theory: Coding Theorems for Discrete Memoryless System (http://www.amazon.com/Information-Theory-Theorems-Discrete-Memoryless/dp/0521196817/ref=sr_1_1?s=books&ie=UTF8&qid=1333586246&sr=1-1) by Imre Czsiar and Janos Korner
A great text from the Hungarian school of information theory.
Entropy and Information Theory (http://ee.stanford.edu/~gray/it.html) by R.M. Gray (one of the leading living information theorists - freely available online)
Linear Algebra Done Right (http://www.amazon.com/gp/product/0387982582/) by Axler
Kronecker is said to have remarked, "God made the natural numbers; all else is the work of man." Number theory is the study of the natural numbers.
A Classical Introduction to Modern Number Theory (http://www.amazon.com/gp/product/038797329X/) by Ireland and Rosen
Algebraic Number Theory (http://amzn.com/0950273422/) by Cassels and Fröhlich
Introduction to Lattices and Order (http://www.amazon.com/gp/product/0521784514/sr=1-2/qid=1139005389/ref=sr_1_2/002-0300919-4226435?%5Fencoding=UTF8) by Davey and Priestley
This is not the place for general physics book recommendations, but there are a few textbooks written to introduce the physicist to relevant mathematics.
Mathematical Methods of Classical Mechanics (http://www.amazon.com/gp/product/0387968903/) by Arnold
An excellent mathematical methods book for physicists is Mary Boas' Mathematical Methods in the Physical Sciences (http://www.amazon.com/gp/product/0471198269/).
Mathematical Methods for Physicists (http://www.amazon.com/Mathematical-Methods-Physicists-Sixth-Comprehensive/dp/0120598760/ref=dp_ob_title_bk) by Arfken & Weber
Presents a good overview of undergraduate mathematics for physics, and a good reference.
Principles of Mathematical Analysis (http://www.amazon.com/gp/product/007054235X/) by Rudin
Principles is also known as "Baby Rudin." It covers all the basics of single variable theory, then goes on to differential forms and Stokes' theorem, and then Lebesgue integration.
Real and Complex Analysis (http://www.amazon.com/gp/product/0070542341/) by Rudin
Introduction to Integration (http://www.amazon.com/gp/product/0198501234/) by Priestley
Elementary Classical Analysis (http://www.amazon.com/Elementary-Classical-Analysis-Jerrold-Marsden/dp/0716721058/ref=sr_1_1?ie=UTF8&s=books&qid=1272769447&sr=1-1) by Marsden & Hoffman
A text which the author states is at a similar level to Baby Rudin. Hints to problems at back. Older printings have significant errata. Extensive Bibliography, and applications to physics and engineering towards the end.
Analysis in Euclidean Space (http://www.amazon.com/Analysis-Euclidean-Space-Kenneth-Hoffman/dp/0486458040/ref=sr_1_3?ie=UTF8&s=books&qid=1272769523&sr=1-3) by Hoffman
A Dover book which concentrates mostly on the real numbers instead of general metric space cases for the text. Good problems.
Elementary Analysis: The Theory of Calculus (http://www.amazon.com/Elementary-Analysis-Calculus-Undergraduate-Mathematics/dp/1441928111/ref=sr_1_1?s=books&ie=UTF8&qid=1333586408&sr=1-1) by Kenneth Ross
A very gentle introduction to real analysis (and also quite a cheap text); Make sure to get a fairly new printing (since they are typeset in LaTeX).
Representation theory studies how groups can act on vector spaces. This carries a great deal of information about the group itself. See also: algebra (alperin & bell).
Character Theory of Finite Groups (http://www.amazon.com/gp/product/0486680142/) by Isaacs
Representation Theory: A First Course (http://www.amazon.com/gp/product/0387974954/) by Fulton and Harris
Riemannian geometry, also known as differential geometry, is the study of Riemannian manifolds — manifolds equipped with a notion of length.
Riemannian Geometry (http://www.amazon.com/gp/product/3540204938/) by Gallot, Hulin and Lafontaine
Heat Kernels and Dirac Operators (http://www.amazon.com/gp/product/3540200622/) by Berline, Getzler and Vergne
Set Theory and Logic (http://www.amazon.com/gp/product/0486638294/) by Stoll
Topos Theory (http://www.amazon.com/gp/product/0123878500/) by Johnstone
CAT(0) spaces, yea!
Metric Spaces of Non-Positive Curvature (http://www.amazon.com/gp/product/3540643249/) by Bridson and Haefliger
Theory of Computation
Theory of computation is the branch of computer science that studies which problems are solvable by computers and how solvable problems can be solved efficiently. It is divided into two major areas: computability theory (which deals with which problems are solvable) and complexity theory (which deals more with efficiency of solutions).
Introduction to the Theory of Computation (http://www.amazon.com/gp/product/0534950973/sr=1-1/qid=1139001837/ref=pd_bbs_1/002-0300919-4226435?%5Fencoding=UTF8) by Sipser
Sipser's book is a slow, gentle, and intuitive introduction to the subject.
Elements of the Theory of Computation (http://www.amazon.com/gp/product/0132624788/sr=1-3/qid=1139001705/ref=pd_bbs_3/002-0300919-4226435?%5Fencoding=UTF8) by Lewis and Papadimitriou
Lewis and Papadimitriou is a denser and somewhat more advanced text, but it covers more material and gives a clearer and more rigorous explanation of some of the topics covered in Sipser.
Introduction to Automata Theory, Languages, and Computation (http://www.amazon.com/gp/product/0201441241/) by Hopcroft and Ullman.
General Topology (http://www.amazon.com/gp/product/0387901256/) by Kelley
Topology and Geometry (http://www.amazon.com/gp/product/0387979263/) by Bredon
Topology (http://www.amazon.com/gp/product/0131816292/) by Munkres
The first edition forms essentially the first half of the second edition (where the second half forms a basic introduction to algebraic topology).
General Topology (http://www.amazon.com/gp/product/B0006D0K3S/) by Engelking
Notes by Allen Hatcher (http://www.math.cornell.edu/~hatcher/Top/Topdownloads.html)
A Concise Course in Algebraic Topology (http://www.math.uchicago.edu/~may/BOOKSMaster.html) by Peter May (really concise)
See also this recommended list by Allen Hatcher (http://www.math.cornell.edu/~hatcher/Other/otherbooks.html)
Computional and Engineering Mathematics
Applied Numerical Linear Algebra (http://www.amazon.com/gp/product/0898713897/) - James W Demell
Derives all the most common algorithms for dealing with eigenvalue problems and large matrix problems, as well as linear least squares. From QR-iteration to Krylow spaces to the multigrid method. Includes many exercises, easy-hard.
Computational Differential Equations (http://www.amazon.com/gp/product/0521567386/) - Kenneth Ericsson, Donald Estep, Peter Hansbo, Claes Johnson
Simpler book on PDEs and solving them with FEM, although some mathematical maturity is recommended. Exercises are rather hard without the right maturity. First part has focus on solving ODE/BVP, the second part extends to PDE.
Partial Differential Equations with Numerical Methods (http://www.amazon.com/gp/product/3540017720/) - Stig Larsson, Vidar Thomee
Condensed book on PDEs and how to solve them using FEM and to a small extent finite differences. Several nice inequalities shown in the context of apriori and a posteriori error estimation. Quite hard exercises; a quick tour in applied functional analysis is recommended, or at least having read CDE, see above.
Finite Markov Chains and Algorithmic Applications (http://www.amazon.com/gp/product/0521890012/) - Olle Häggström
Covers simulated annealing, metropolis chains and more complicated methods like sandwitching. These are techniques for solving problems with many dimensions in contexts like optimization. Requires no prior knowledge in markov chains. Easy exercises, analytical and programming.
Computational Physics (http://www.amazon.com/gp/product/0521575885/) - J M Thijssen
Has nice mathematical content despite the name. Covers FEM, monte carlo etc and gives applications in higher physics, such as computing electron orbitals or studying diffusion. Programming exercises.
Statistical digital signal processing and Modeling (http://www.amazon.com/gp/product/0471594318/)
Covers pretty much everything one need to know about basic 1D signal analysis. Deals with everything from AR-processes to Kalman filtering. Rather theoretical.
On Growth, Form and Computation (http://www.amazon.com/gp/product/0124287654/)
Recent ideas from morphology, includes chapters written by many leading researchers (this makes it a bit messy though). Deals mainly with evolution, how to simulate it instances of it, and the relation between symmetries and complex systems in nature.
Computational Cell Biology (http://www.amazon.com/gp/product/0387953698/) - Christopher P Fall, Eric S Marland, John M Wagner, John J Tyson
Unless someone has something better; describes cell systems, models and how to analyze them mathematically. Unlike other texts, this is written by and for mathematicians so it is rather easy to pick up without prior courses in biology. Rather easy but lengthy exercises.