Integration of Differential Forms
From Theodore Frankel, The Geometry of Physics - "Exterior differential forms (differential forms, p forms) are natural objects appearing as integrands of line, surface, and volume integrals as well as the n-dimensional generalizations require in, for example, Hamiltonian mechanics, relativity, and string theories. One does not integrate vectors; one integrates forms. If a line integral of a vector occurs in a problem, then usually a deeper look at the situation will show that the vector in question was in fact a covector; that is a 1-form! There are times when one needs to compute the arc length of a curve, but, usually, it is completely irrelevant to either the computation or the concept of a line integral! Line (and, as we shall see, surface) integrals are independent of any metric notions in space. This is one case where the usual elementary treatment given in many calculus texts is harmful and misleading and should have been discarded long ago." Keep learning Differential Geometry! My aim is to read 3 chapters a day of Frankel's Geometry of Physics. Watch out for my #indiegogo campaign!
