In mathematics, the discussion of vector fields on spheres was a classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras.
Specifically, the question is how many linearly independent vector fields can be constructed on a sphere in N-dimensional Euclidean space. A definitive answer was made in 1962 by Frank Adams. It was already known, by direct construction using Clifford algebras, that there were at least ρ(N) such fields (see definition below). Adams applied homotopy theory to prove that no more independent vector fields could be found.
Read more about Vector Fields On Spheres: Technical Details, Radon–Hurwitz Numbers
Famous quotes containing the words fields and/or spheres:
“Come up from the fields father, here’s a letter from our Pete,
And come to the front door mother, here’s a letter from thy dear
son.”
—Walt Whitman (1819–1892)
“The world has already learned that woman has other virtues than meekness, patience, humility and endurance. She possesses courage above all fear, and a will that knows no obstacles; and when these are called forth by some great emergency, false modesty is trampled in the dust, and spheres are scattered to the winds.”
—A. Holley, U.S. women’s magazine contributor. The Lily, p. 38 (May 1852)