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A 1-form w is said to be exact in a region R if there is a function f that is defined and of class C^1 (i.e., is once continuously differentiable in R) and such that df=w.

Donaldson (1983) showed there exists an exotic smooth differential structure on R^4. Donaldson's result has been extended to there being precisely a continuum of ...

The kth exterior power of an element alpha in an exterior algebra LambdaV is given by the wedge product of alpha with itself k times. Note that if alpha has odd degree, then ...

If it is possible to transform a coordinate system to a form where the metric elements g_(munu) are constants independent of x^mu, then the space is flat.

"Fluxion" is the term for derivative in Newton's calculus, generally denoted with a raised dot, e.g., f^.. The "d-ism" of Leibniz's df/dt eventually won the notation battle ...

A connection defined on a smooth algebraic variety defined over the complex numbers.

A geodesic mapping f:M->N between two Riemannian manifolds is a diffeomorphism sending geodesics of M into geodesics of N, whose inverse also sends geodesics to geodesics ...

A metric on a bunch of segments with a common endpoint O, which defines the distance between two points P_1 and P_2 as the length of the shortest path connecting them inside ...

Hilbert-Schmidt theory is the study of linear integral equations of the Fredholm type with symmetric integral kernels K(x,t)=K(t,x).

On an oriented n-dimensional Riemannian manifold, the Hodge star is a linear function which converts alternating differential k-forms to alternating (n-k)-forms. If w is an ...