Lipschitz classes are important to determing how bijectively this statement holds numerically.  This statement relates an analytic functional and an algebraic operator, the latter of which is related to the formal de Rham Cohomology. It can be extended to parametrization along a curve in higher-dimensional space, which in t
Lipschitz classes are important to determing how bijectively this statement holds numerically.  This statement relates an analytic functional and an algebraic operator, the latter of which is related to the formal de Rham Cohomology. It can be extended to parametrization along a curve in higher-dimensional space, which in the case of irrotational vector fields uses Poincare's theorem to yield the gradient theorem, and it can be extended to complex functions as long as they remain continuous in their domain.  One proof of this result, which was established independently by two mathematicians, involves taking a continuous function, applying the Mean Value Theorem to successive divisions of some interval, and then noting that sum is bounded above and below by an object named for Reimann.  For 10 points, name this pair of theorems that relate the two important branches in its namesake field of math, the definite integral and the derivative.
Lipschitz classes are important to determing how bijectively this statement holds numerically.  
{Fundamental Theorem}s {of Calculus} [do not penalize for singular/plural mistakes, accept {Second} or {First }in front of {Fundamental} because textbooks just can't decide on whether they are really one theorem or two]