This is in contrast to the unsigned deﬁnite integral R [a,b] f(x) dx, since the set [a,b] of numbers between a and b is exactly the same as the set of numbers between b and a. They are dual to vectors, so they measure them which can be visualized with planes the vectors pierce. People have defined differential forms as "the things you integrate", so don't be alarmed. Differential forms are supposed to be giving meaning to the ##dt##, but with this viewpoint I am just circling back to "it's the variable to integrate over". This book explains and helps readers to develop geometric intuition as it relates to differential forms. Active 3 months ago. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. In the following, we provide a primer on di erential forms with an emphasis on their relevance in modern classical mechanics which tries to convey the intuition … For this reason, when one is studying cohomology with differential forms, you are actually studying (by one of these theorems that says all cohomology theories that satisfy modest axioms are isomorphic!) A 0-form is just a function. In this particular book, the authors motivate the anti-symmetry condition by properties of determinants and Jacobian's for change of variables in integration. For 1-forms, you can get some intuition for exterior differentiation from how it shows up in Frobenius's theorem which states that a distribution D is integrable if and only if the ideal of differential forms that are annihilated by it is closed under exterior differentiation: Di erential forms are ubiquitous in modern mathematical physics and their relevance for computations has increasingly been realized. studying the geometry of a dual triangulation of your space. Chapter 7 of my online book Classical and quantum mechanics via Lie algebras derives in 17 pages (pp. 161-177) the main concepts of equilibrium thermodynamics in a physically elementary and mathematically rigorous form. Loosely speaking, to avoid giving an overly formal definition: “things you integrate.” A 1-form is something you integrate over a line, a 2-form over an area, etc. 11 \$\begingroup\$ I think I understood 1-forms fairly well with the help of these two sources. 1 1-forms 1.1 1-forms A di erential 1-form (or simply a di erential or a 1-form) on an open subset of R2 is an expression F(x;y)dx+G(x;y)dywhere F;Gare R-valued functions on the open set. Ask Question Asked 1 year, 8 months ago. Building Intuition for Differential forms, exterior derivative, wedge. It includes over 250 figures to aid understanding and enable readers to visualize the concepts being discussed. This book explains and helps readers to develop geometric intuition as it relates to differential forms. This gives you intuition for why they should be similar, but can often be different. But I was wondering if there are other ways to think about why differential forms should commute anti-symmetrically which might provide some more intuition on just why this "miracle" works. A Visual Introduction to Differential Forms and Calculus on Manifolds Fortney, J.P. DIFFERENTIAL FORMS AND INTEGRATION 3 Thus if we reverse a path from a to b to form a path from b to a, the sign of the integral changes. Viewed 914 times 11.