Fundamental theorem of calculus Totally Explained

Everything about The Fundamental Theorem Of Calculus totally explained

The fundamental theorem of calculus specifies the relationship between the two central operations of calculus, differentiation and integration.
   The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration can be reversed by a differentiation.
   The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals.
   The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638-1675). Isaac Barrow proved the first completely general version of the theorem, while Barrow's student Isaac Newton (1643–1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities.
   The Fundamental Theorem of Calculus is sometimes also called Leibniz's Fundamental Theorem of Calculus or Torricelli-Barrow Fundamental Theorem of Integral Calculus.

Intuition

Intuitively, the theorem simply states that the sum of infinitesimal changes in a quantity over time (or some other quantity) add up to the net change in the quantity.
To comprehend this statement, we'll start with an example. Suppose a particle travels in a straight line with its position given by x(t) where t is time and x(t) means that x is a function of t. The derivative of this function is equal to the infinitesimal change in quantity, dx, per infinitesimal change in time, dt (of course, the derivative itself is dependent on time). Let us define this change in distance per change in time as the speed v of the particle. In Leibniz's notation:
ť

frac!,

is the exterior derivative, which is defined using the manifold structure only.
The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold on which the form

omega

is defined.

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