# Different ways of treating vector calculus?

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?

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This text is one of the first to treat vector calculus using differential forms in place of vector fields and other outdated techniques.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

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