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I learned that there are different ways of treating vector calculus: vector fields and differential forms, if I understand correctly. The former is used in calculus, and the latter is in differential geometry. My memory of calculus is vague on the vector calculus part and I have limit knowledge about differential geometry, so I was wondering how these two ways are different and related in a brief? There is no need to refrain from using some terminology in reply, and I can look them up if I am not familiar.

A side question: is multivariate calculus synonym of vector calculus?

Thanks for your input!

PS: The book description of Differential Forms: Integration on Manifolds and Stokes's Theorem by Steven H. Weintraub is from which I learned the above:

Book Description:

This text is one of the first to treat vector calculus using differential forms in place of vector fields and other outdated techniques. Geared towards students taking courses in multivariable calculus, this innovative book aims to make the subject more readily understandable. Differential forms unify and simplify the subject of multivariable calculus, and students who learn the subject as it is presented in this book should come away with a better conceptual understanding of it than those who learn using conventional methods.

  • Treats vector calculus using differential forms
  • Presents a very concrete introduction to differential forms
  • Develops Stokess theorem in an easily understandable way
  • Gives well-supported, carefully stated, and thoroughly explained definitions and theorems.
  • Provides glimpses of further topics to entice the interested student
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MV Calculus and "Vector Calculus" generally both mean the same thing: Analysis performed in $\mathbb{R}^n$. In either case, differential forms and the associated abstract machinery (manifolds, exterior algebra, etc) is a (vast?) generalization of "Vector Calculus". The results in vector calculus can be obtained by using $\mathbb{R}^3$ as the manifold and there is a very famous set of isomorphisms that relate the exterior derivative to the classical vector calculus operations DIV/GRAD/CURL. See the first pages of Bott and Tu's Differential Forms in Algebraic topology for details on this. –  ItsNotObvious Dec 2 '11 at 14:19
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Trying to put it simply, I'll look at n - dimensional Euclidean space first. Then, differential forms (1 forms to be more precise) are dual to vector fields, that is, a 1 - form $\omega$ is nothing but a linear function on the space of tangent vectors in p.

For differential forms the notion of exterior product or wedge product is rather important, n-forms in n-dimensional space corrsponding to a multiple of the determinant function $\det$. This makes them volume forms and gives rise to a an integral (as are one forms when restricted to the tangent space of curves) for which particularly nice theorems (like Stokes theorem) are rather elegantly to formulate. k - forms are then natural candidates for volume form on k-dimensional submanifolds

This notion may be extended to vector bundles on manifolds, the precise defnitions are a bit involved, though. In differential geometry, both approaches (vector fields and forms) are used, and often combined. You can do tensor analysis and look at exterior algebras on these objects by first doing it locally on the product of an open set and a vector space and then gluing thing together using an atlas. The term used in that general case is 'tensor analysis'. A detailed discussion of several approaches to work with these concepts is found in Michael Spivaks 'Comprehensive Introduction to differential geometry', the first and second volume focus on these topics. In particular it explains how to 'translate' between the various concepts. (There may be newer books, I worked on that about 20 years ago, and I still love the humor in Spivaks books. :-)

In lectures or even books on differential geometry you will usually see a focus one of these approaches. This is due to the fact that very often the linear algebra involved is often no longer taught in basics lectures and has to be presented, too. Then there is no time left to cover the dual view and to show how the concepts relate to each other.

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