Calculus on Manifolds
Encyclopedia
Michael Spivak
's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965, ISBN 0-8053-9021-9) is a short text treating analysis in several variables in Euclidean space
s and on differentiable manifold
s. The textbook is famously used in the honors Analysis II courses at The University of Chicago, Harvard, Princeton
, MIT, and Toronto
, and the Vector Calculus course at Auburn University
. The book develops the classical theorems of advanced calculus, those of Green
, Gauss, and Stokes
, in the language of differential form
s and in the context of differentiable manifold
s embedded in Euclidean space
. Calculus on Manifolds aims to present the topics of vector analysis in the manner that they are seen by a working mathematician, yet simply and selectively enough to be understood by strong undergraduate students.
Another textbook often used in undergraduate courses to supplement Spivak's book is the text Analysis on Manifolds (1997, ISBN 978-0-20131596-7) by James Munkres
.
The book is affectionately referred to as "Baby Spivak" due to its diminutive size (8 x 5.5 x 0.6 inches (slightly smaller than A5 paper), soft cover, 160 pp.), in contrast to his much larger Calculus (10 x 9.3 x 1.5 inches, hard cover, 680 pp.).
The cover of Calculus on Manifolds features a copy of the original publication of Stokes' theorem
as it was written in a letter by Lord Kelvin to Sir George Stokes
.
Michael Spivak
Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. He is the author of the five-volume Comprehensive Introduction to Differential Geometry. He received a Ph.D...
's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965, ISBN 0-8053-9021-9) is a short text treating analysis in several variables in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
s and on differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s. The textbook is famously used in the honors Analysis II courses at The University of Chicago, Harvard, Princeton
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, MIT, and Toronto
University of Toronto
The University of Toronto is a public research university in Toronto, Ontario, Canada, situated on the grounds that surround Queen's Park. It was founded by royal charter in 1827 as King's College, the first institution of higher learning in Upper Canada...
, and the Vector Calculus course at Auburn University
Auburn University
Auburn University is a public university located in Auburn, Alabama, United States. With more than 25,000 students and 1,200 faculty members, it is one of the largest universities in the state. Auburn was chartered on February 7, 1856, as the East Alabama Male College, a private liberal arts...
. The book develops the classical theorems of advanced calculus, those of Green
Green's theorem
In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C...
, Gauss, and Stokes
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
, in the language of differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
s and in the context of differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s embedded in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. Calculus on Manifolds aims to present the topics of vector analysis in the manner that they are seen by a working mathematician, yet simply and selectively enough to be understood by strong undergraduate students.
Another textbook often used in undergraduate courses to supplement Spivak's book is the text Analysis on Manifolds (1997, ISBN 978-0-20131596-7) by James Munkres
James Munkres
James Raymond Munkres is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including Topology , Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology...
.
The book is affectionately referred to as "Baby Spivak" due to its diminutive size (8 x 5.5 x 0.6 inches (slightly smaller than A5 paper), soft cover, 160 pp.), in contrast to his much larger Calculus (10 x 9.3 x 1.5 inches, hard cover, 680 pp.).
The cover of Calculus on Manifolds features a copy of the original publication of Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
as it was written in a letter by Lord Kelvin to Sir George Stokes
George Stokes
George Stokes is the name of:* Sir George Stokes, 1st Baronet , Irish mathematician and physicist* George Henry Stokes , Member of Canadian Parliament* George Thomas Stokes , Irish historian...
.
External links
- Calculus on Manifolds at the publisher 5th ed. (1971)