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What are differential forms?
Is this what I'm going to learn about in my upcoming differential geometry class?
Is this what I'm going to learn about in my upcoming differential geometry class?
A classical course on differential geometry -- which includes many introductory courses -- may not cover and use differential forms at all.
quetzalcoatl9 said:The short (and least satisfying) answer is that an n-form is an antisymmetric tensor of rank (0,n) that enjoys some nice symmetry properties - these properties essentially make the indices of the tensor "disappear". They can be thought of as functions that take tangent vectors as input They are also well defined across something called "pullback", something that is not true of tensors in general, so in effect they transcendent tensors. They also can provide topological information.
... then I can recommend the book by Weintraub (sp? it may be Weintraube, I forget, he is a professor in Louisianna) called "Differential Forms: A Complement to Vector Calculus" or something like that.
mathwonk said:when is someone going to reveal that there is an entire thread devoted to this topic, called "a geometric approach to differential forms" by david bachman, just below here, with zillions of entries and a free book?
quetzalcoatl9 said:Try doing a search on this forum and you will see previous discussions.<snip>
Differential forms are a mathematical tool used in differential geometry to describe geometric objects and their properties in a coordinate-independent way. They allow for a deeper understanding of the underlying structure of a manifold and can be used to solve various problems in physics and engineering.
Differential forms are multilinear and antisymmetric, meaning they are sensitive to the orientation and ordering of basis vectors. This makes them well-suited for describing geometric properties such as area, volume, and orientation on a manifold without reference to a specific coordinate system.
Differential forms have a wide range of applications in various fields including physics, engineering, and computer graphics. They are used to solve problems in electromagnetism, fluid dynamics, and general relativity, among others. They are also essential in the development of numerical methods for solving partial differential equations.
Differential forms can be thought of as a generalization of vector calculus to higher dimensions. They provide a rigorous mathematical framework for solving differential equations on manifolds and are essential in the study of differential geometry, which is a fundamental tool in many areas of mathematics and science.
There are many books and online resources available for learning about differential forms and differential geometry. Some popular texts include "Differential Forms in Algebraic Topology" by Raoul Bott and Loring Tu, "Differential Forms and Applications" by Manfredo P. do Carmo, and "Differential Geometry of Curves and Surfaces" by Kristopher Tapp. Online resources such as lectures, notes, and video tutorials can also be found on various websites and platforms such as YouTube and Coursera.