Editing Affine connection (section)

===Motivation from tensor calculus===
{{see also|Covariant derivative}}

[[File:Affine connection example.svg|thumbnail|Historically, people used the covariant derivative (or Levi-Civita connection given by the metric) to describe the variation rate of a vector along the direction of another vector. Here on the punctured 2-dimensional Euclidean space, the blue vector field {{mvar|X}} sends the [[one-form]] {{math|d''r''}} to 0.07 everywhere. The red vector field {{mvar|Y}} sends the one-form {{math|''r''d''θ''}} to {{Math|0.5''r''}} everywhere. Endorsed by the metric {{math|d''s''<sup>2</sup> {{=}} d''r''<sup>2</sup> + ''r''<sup>2</sup>d''θ''<sup>2</sup>}}, the Levi-Civita connection {{math|∇<sub>''Y''</sub>''X''}} is 0 everywhere, indicating {{mvar|X}} has no change along {{mvar|Y}}. In other words, {{mvar|X}} [[parallel transport]]s along each [[concentric]] circle. {{math|1=∇<sub>''X''</sub>''Y'' = ''Y''/''r''}} everywhere, which sends {{math|''r''d''θ''}} to 0.5 everywhere, implying {{mvar|Y}} has a "constant" changing rate on the radial direction.]]
The second motivation for affine connections comes from the notion of a [[covariant derivative]] of vector fields. Before the advent of coordinate-independent methods, it was necessary to work with vector fields by [[embedding]] their respective [[Euclidean vector]]s into an [[Atlas (topology)|atlas]]. These components can be differentiated, but the derivatives do not transform in a manageable way under changes of coordinates.{{citation needed|date=June 2016}} Correction terms were introduced by [[Elwin Bruno Christoffel]] (following ideas of [[Bernhard Riemann]]) in the 1870s so that the (corrected) derivative of one vector field along another transformed [[covariant transformation|covariantly]] under coordinate transformations &mdash; these correction terms subsequently came to be known as [[Christoffel symbol]]s.

This idea was developed into the theory of ''absolute differential calculus'' (now known as [[tensor calculus]]) by [[Gregorio Ricci-Curbastro]] and his student [[Tullio Levi-Civita]] between 1880 and the turn of the 20th century.

Tensor calculus really came to life, however, with the advent of [[Albert Einstein]]'s theory of [[general relativity]] in 1915. A few years after this, Levi-Civita formalized the unique connection associated to a Riemannian metric, now known as the [[Levi-Civita connection]]. More general affine connections were then studied around 1920, by [[Hermann Weyl]],<ref>{{Harvnb|Weyl|1918}}, 5 editions to 1922.</ref> who developed a detailed mathematical foundation for general relativity, and [[Élie Cartan]],<ref name="Cartan-affine">{{Harvnb|Cartan|1923}}.</ref> who made the link with the geometrical ideas coming from surface theory.