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In undergraduate study, students are taught that definite integral involves Riemann or Lebesque partial sums with respect to some measure, so an integral is written as ∫f(x) dx. From the outset, dthe integrand f(x) and the integration measure dx within the definite integral are usually regarded as separate entities. However, this viewpoint does not fully reflect the fact that we can change the integration variable without affecting the result of the integral:
\[ \int_a^b f(x)\,{\text d}x = \int_{\alpha}^{\beta} f(\xi )\,{\text d}\xi , \qquad x = \phi (\xi ) . \]
The notion of a differential form arises when expressing the integral in a way that explicitly treats all possible choices of integration variable on an equal footing. The key conceptual step is to combine f(x) and dx into the same object. We introduce the differential 1-form γ = f(x) dx and write the integral as ∫Iγ, where I is equivalent to the interval 𝑎 ≤ x ≤ b.

Differential Forms

Élie Cartan
Differential forms are part of the field of differential geometry, influenced by linear algebra. Although the notion of a differential is quite old, the initial attempt at an algebraic organization of differential forms is usually credited to Élie Cartan (1869--1951) with reference to his 1899 paper.
A differential one-form (or simply 1-form) on an open subset of ℝ³ is an expression F(x, y, z) dx + G(x, y, z) dy + H(x, y, z) dz, where F, G, H are ℝ-valued functions on the open set. If f(x, y, z) is a 𝒞¹ function (or 0-form) on this set, then its total differential is \[ {\text d}f = \frac{\partial f}{\partial x}\,{\text d} x + \frac{\partial f}{\partial y}\,{\text d} y + \frac{\partial f}{\partial z}\,{\text d} z , \]


  1. Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International
  2. Cartan, E., "Sur certaines expressions différentielles et le problème de Pfaff", Annales Scientifiques de l'École Normale Supérieure, 16: 239–332, doi:10.24033/asens.467