Customer Reviews (1)

"A new type of differentiation"
This is a fascinating history of partial differentiation. Interestingly, partial differentiation was invented to solve a problem that had been solved already, because the existing solution was considered unsatisfactory in a sort of half mathematical and half epistemological sense. The problem in question is a spin-off of the famous brachistochrone problem, which asks for the path of quickest descent from a given point A to a given point B. The spin-off problem is to find the path of quickest descent from A to a given vertical line L.

One way of solving this problem is by considering synchrones. Consider the family of all brachistochrone curves passing through A (obtained by letting B vary). Then imagine releasing a particle along each curve and freezing them all after a fixed time. The positions of the particles determine a synchrone curve.

(The motivation for considering these curves was partly optical. Johann Bernoulli of course famously solved the brachistochrone problem by exploiting the analogy between the gravitational descent of a particle and the path of a light ray through a substance consisting of thin horizontal layers of variable density. In this analogy, the synchrones correspond to Huygensian wavefronts.)

The problem of finding the path of quickest descent from A to a given vertical line L is now "solved" in a sense: simply find the synchrone to which L is tangent; then that point of tangency will obviously be the point on L that is quickest to reach from A, and the quickest path is consequently the brachistochrone passing through this point.

This solution is problematic, however, for it is not clear to what extent the synchrones should be considered "known". It is not difficult to construct points on the synchrones: since we know how fast the particle is travelling at any given point, to find its position after a given time reduces to an arc-length problem. Thus the point at which the synchrone for a particular t intersects a given brachistochrone is implicitly known as whatever upper bound makes the brachistochrone's arc-length integral come out equal to a given value. But this integral is transcendental, so we cannot solve for the upper bound and obtain an explicit formula for the synchrone.

Thus the synchrones are known only by a pointwise construction that requires the rectification of a different curve for each of its points. This mode of construction, Johann Bernoulli complained to Leibniz (p. 41), does not allow one to find tangents, as the above "solution" requires.

It was in response to this challenge that Leibniz invented partial differentiation: the tangent to the synchrone can be found by differentiating "from curve to curve", i.e. with respect to the parameter that determines which brachistochrone we are on. Differentiation with respect to the parameter can be moved under the integral sign, and in this way the need for solving for the upper bound can be circumvented.

"Certainly nothing more elegant exists or can be invented," said Bernoulli, and proposed that "we can torment the Geometers" with problems based on this new method. "They would marvel about their own infirmity the more, since this type of problem seems so easy and seems to be taken only from the direct method of tangents [i.e., it seems to involve only differentiation]." (p. 49)

But despite early success in some isolated examples, the scope of the method of differentiation with respect to the parameter proved disappointing: it "excels in curiosity rather than utility when applied to examples" (p. 91), admitted Bernoulli about his attempts to solve orthogonal-trajectory problems by this method when he finally published it some 23 years later. ... Read more

Product Description

This collection covers the state of the art in automatic differentiation theory and practice. Practitioners and students will learn about advances in automatic differentiation techniques and strategies for the implementation of robust and powerful tools. Computational scientists and engineers will benefit from the discussion of applications, which provide insight into effective strategies for using automatic differentiation for design optimization, sensitivity analysis, and uncertainty quantification.

Product Description

This collection covers advances in automatic differentiation theory and practice. Computer scientists and mathematicians will learn about recent developments in automatic differentiation theory as well as mechanisms for the construction of robust and powerful automatic differentiation tools. Computational scientists and engineers will benefit from the discussion of various applications, which provide insight into effective strategies for using automatic differentiation for inverse problems and design optimization.

Product Description
This textbook and treatise begins with classical real variables, develops the Lebesgue theory abstractly and for Euclidean space, and analyzes the structure of measures. The authors' vision of modern real analysis is seen in their fascinating historical commentary and perspectives with other fields. There are comprehensive treatments of the role of absolute continuity, the evolution of the Riesz representation theorem to Radon measures and distribution theory, weak convergence of measures and the Dieudonné-Grothendieck theorem, modern differentiation theory, fractals and self-similarity, rearrangements and maximal functions, and surface and Hausdorff measures. There are hundreds of illuminating exercises, and extensive, focused appendices on functional and Fourier analysis. The presentation is ideal for the classroom, self-study, or professional reference. ... Read more

Product Description
This book introduces students to vector analysis, a concise way of presenting certain kinds of equations and a natural aid for forming mental pictures of physical and geometrical ideas. Students of the physical sciences and of physics, mechanics, electromagnetic theory, aerodynamics and a number of other fields will find this a rewarding and practical treatment of vector analysis. Key points are made memorable with the hundreds of problems with step-by-step solutions, and many review questions with answers. ... Read more

Customer Reviews (17)

Not satisfied
Papers were yellow as the book was a 1993 edition. It was not mentioned in the ad. I expected a more recent edition.

Stop Reading. Buy It NOW
If you are reading this review, then you should definitely own this book. Express ship it and work the problems immediately. This book is gold. Like another reviewer says, I would rate this more than five stars, if possible. If I had this book years ago, my life would have been so much easier.

For some reason, vector analysis is generally left to self study or briefly summarized for physics majors in the USA. This is very tragic. Do yourself a favor and get this book ASAP. You will be so far ahead of the pack that you will wonder what everyone else thinks is so difficult about Electromagnetism and Mechanics. Don't wait. Buy it now.

Excellent Course Analysis With Physically Myopic Flaw
Different schools of thought have different checks and balanaces.These checks and balances take the forms of rules and norms with their own judgements of right and wrong.Probably the human mind can only take so many rules and norms at any one time; each school of thought somehow subconsciously understands this.

Still, time and the wisdom that comes with it should make what I am saying obvious to all.And perhaps my point is indeed obvious to Prof. Spiegel.If so, he should note in his book somehow.Either in the discussion or in the problems.

My main point and problem with this presentation on Vector Analysis is that the definition of vector is graphically inadequate.Pictorally speaking, position, separation, and displacement vectors have features that many other vectors don't.Such vectors must be treated somewhat differently than velocity, acceleration, and force vectors.Now velocity and acceleration vectors may have different features and need to be treated differently, but my point has already been made in identifying the existence of unique features of location vectors.

Murray R. Spiegel has proven himself again and again in book after book as a great master of many topics as mathematical subjects.From Complex Variables to Vector Analysis.

Still, he makes no mention that mathematicians take great liberties when they define concepts.And vector as "a quantity with magnitude and direction" or, worse, as "a vertical column or horizontal row of numbers"
are exact examples of such liberties and their problematic consequences.

Bottom Line:Great presentation of the symbols and the symbolic logic of vectors.Conceptually, Graphically, and Physically, speaking the analysis (presentation) is quite weak and quite misleading.

Easy written, lot of examples
The book offers a good introduction to vector analysis for the undergraduate student. Quite simple, lot of practical examples. It is a good supplement to a more theoretical textbook. Also the price is fine.

You won't be disappointed
This book is worth a whole lot more than the paltry sum they're charging. You really can't go wrong with any mathematics or physics material authored by Murray Spiegel but this text is in a class by itself. Even though it's a Schaum's "Outline", you'll find that it's actually a first rate textbook. I'd say it's an outstanding learning tool given the fact that I found myself actually enjoying working through it! ... Read more

Product Description
A rigorous introduction to calculus in vector spaces

The concepts and theorems of advanced calculus combined with related computational methods are essential to understanding nearly all areas of quantitative science. Analysis in Vector Spaces presents the central results of this classic subject through rigorous arguments, discussions, and examples. The book aims to cultivate not only knowledge of the major theoretical results, but also the geometric intuition needed for both mathematical problem-solving and modeling in the formal sciences.

The authors begin with an outline of key concepts, terminology, and notation and also provide a basic introduction to set theory, the properties of real numbers, and a review of linear algebra. An elegant approach to eigenvector problems and the spectral theorem sets the stage for later results on volume and integration. Subsequent chapters present the major results of differential and integral calculus of several variables as well as the theory of manifolds. Additional topical coverage includes:

• Sets and functions

• Real numbers

• Vector functions

• Normed vector spaces

• First- and higher-order derivatives

• Diffeomorphisms and manifolds

• Multiple integrals

• Integration on manifolds

• Stokes' theorem

• Basic point set topology

Numerous examples and exercises are provided in each chapter to reinforce new concepts and to illustrate how results can be applied to additional problems. Furthermore, proofs and examples are presented in a clear style that emphasizes the underlying intuitive ideas. Counterexamples are provided throughout the book to warn against possible mistakes, and extensive appendices outline the construction of real numbers, include a fundamental result about dimension, and present general results about determinants.

Assuming only a fundamental understanding of linear algebra and single variable calculus, Analysis in Vector Spaces is an excellent book for a second course in analysis for mathematics, physics, computer science, and engineering majors at the undergraduate and graduate levels. It also serves as a valuable reference for further study in any discipline that requires a firm understanding of mathematical techniques and concepts. ... Read more