Hermann Günter Grassmann

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Born: 15 April 1809 in Stettin, Prussia (now Szczecin, Poland)
Died: 26 Sept 1877 in Stettin, Germany (now Szczecin, Poland)

Hermann Grassmann is chiefly remembered for his development of a general calculus for vectors.

Grassmann taught at the Gymnasium in Stettin from 1831 until his death except for two years (1834-1836) when he taught in Berlin.

Grassmann's most important work is Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844) developed the idea of an algebra in which the symbols representing geometric entities such as points, lines and planes, are manipulated using certain rules. He represented subspaces of a space by coordinates leading to point mapping of an algebraic manifold now called the Grassmannian.

Grassmann invented what is now called Exterior Algebra. This was joined to Hamilton's quaternions by Clifford in 1878. Clifford replaced Grassmann's rules

e_p e_p= 0 and e_p e_q= - e_q e_pfor p q

by the rules

e_pe_p= 1 and e_pe_q= - e_qe_p for p q.

Clifford algebras are used today in the theory of quadratic forms and in relativistic quantum mechanics and they appear together
with Grassmann's exterior algebra in differential geometry. See [26].

Grassmann's methods were slow to be adopted but eventually they inspired the work of Elie Cartan and have since been used in studying differential forms and their application to analysis and geometry.

Grassmann wrote on many other subjects, for example electricity, colour, acoustics, linguistics and botany. At the age of 53 he became disappointed with the lack of interest in his mathematical ideas so he turned to Sanskrit studies, another of his interests. His Sanskrit dictionary is still widely used.