Multilinear algebra appears in the study of the mechanical response of materials to stress and strain, involving various moduli of elasticity.
Despite Grassmann's early work in 1844 with his Ausdehnungslehre, which was also republished in 1862, the subject was initially not widely understood, as even ordinary linear algebra posed many challenges at the time.
The concepts of multilinear algebra find applications in certain studies of multivariate calculus and manifolds, particularly concerning the Jacobian matrix.
[3] Following Grassmann, developments in multilinear algebra were made by Victor Schlegel in 1872 with the publication of the first part of his System der Raumlehre[4] and by Elwin Bruno Christoffel.
Notably, significant advancements came through the work of Gregorio Ricci-Curbastro and Tullio Levi-Civita,[5] particularly in the form of absolute differential calculus within multilinear algebra.