On Differential Forms

"Before the development of the theory of relativity it was known the principle of energy and momentum could be expressed in a differential form for the electromagnetic field. The four-dimensional formulation of these principles leads to an important conception, that of the energy tensor, which is important of the further development of the theory of relativity." (Albert Einstein, "The Meaning of Relativity", 1922)

"It was long accepted as a fact that a metrical character could be described by means of a quadratic differential form, but the fact was not clearly understood. Riemann many years ago pointed out that the metrical groundform might, with equal right, essentially, be a homogeneous function of the fourth order in the differentials, or even a function built up in some other way. and that it need not even depend rationally on the differentials. But we dare not stop even at that point." (Hermann Weyl," Space, Time, Matter", 1922)

"In order to get insight in differential geometry, use differential forms and employ their invariance properties." (Eberhard Zeidler, "Quantum Field Theory I: Gauge Theory", 2006)

"In order to define boundary and establish compactness properties, it will be useful to view our rectifiable sets as currents - that is, linear functionals on smooth differential forms (named by analogy with electrical currents with a kind of direction as well as magnitude at a point). The action of an oriented rectifiable set S on a differential form ϕ is given by integrating the form ϕ over the set [...] Currents thus associated with certain rectifiable sets, with integer multiplicities, will be called rectifiable currents. The larger class of normal currents will allow for real multiplicities and smoothing." (Frank Morgan, "Geometric Measure Theory: A Beginner’s Guide" 4th. Ed., 2009)

"An affine space is a space of objects called points such that, after arbitrarily taking a point to play the role of zero, we can establish a one-to-one correspondence between its points and the vectors of a vector space, called its associated vector space. If the vector space is Euclidean, the affine space is called Euclideanspace (i.e. without 'vector' between the words 'Euclidean' and 'space'). The study of affine and Euclidean space is geometry." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"There is a great virtue in calculus of differential forms. It is appropriate for differentiation in general spaces known as differentiable manifolds, which, in general, lack a metric and a connection or rule to compare vectors at different points. We must, however, have a concept of continuity in the space in question, which eliminates automatically discrete sets of points. The set must be such that regions of the same can be represented unequivocally by open sets of n-tuples of real numbers (it could also be complex numbers [...]). The openness of the set has to do with the behavior at the borders of the region." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)

"Vector calculus is horrible for several reasons. One of them is that its curl is based on the vector product. So, we do not have a curl in other dimensions. Another reason is that it uses tangent vectors where it should use differential forms. One more is that it often uses more structure than needed to solve a problem, say a metric structure. Still another one is that one can do so little with it that it has to be complemented with all the other calculi that we also think of replacing with differential forms. The Kähler calculus - based on Clifford algebra of differential forms - replaces tangent vectors and tangent-valued operators with differential forms whose coefficients are respectively functions and operators." (José G Vargas, "Differential Geometry for Physicists and Mathematicians: Moving frames and differential forms from Euclid past Riemann", 2014)