Bernhard Riemann (1826-1866) was a German mathematician. Riemann made
routine use of infinitesimal techniques. Thus, in his famous 1854
essay On the Hypotheses which lie at the Foundations of
Geometry, Riemann speaks of the line as being made up of the
dx, describes dx as `the increments', and speaks of
infinitesimal displacements and of infinitely small quantities
x1 dx2 - x2 dx1,
etc., as well as of infinitely small triangles. In talking about the
`next-order term' in the expansion of the line element on a manifold,
It obviously equal to zero if the manifold in question is flat, i.e., if the square of the line element is reducible to Σ dx2, and can therefore be regarded as the measure of deviation from flatness in this surface direction at this point. When multiplied by -3/4 it becomes equal to the quantity which Privy Councilor Gauss has called the curvature of a surface. (Riemann as translated by Spivak, 1999 volume 2, p. 157).
It is apparent from this passage concerning the Gaussian curvature that Riemann has a pointwise (local) discussion of dx in mind, and not merely an integrand.
What meaning did Riemann assign to dx? (MSE)
More on infinitesimals
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