Infinitesimal calculus MOC

Fundamental theorem of calculus

The fundamental theorem of calculus relates the two main operations of infinitesimal calculus, differentiation and integration, as inverses of each other. Formerly the second part of the theorem states (however still simplified from traditional statements)

Fundamental theorem of calculus, second part Let be a smooth function on . Then calculus

Generalizations

There exist a number of generalizations of the fundamental theorem to different kinds of integrals and derivatives. The most general form, which includes all the below cases as well as the original theorem, is the Generalized Stokes’s theorem which uses the concept of the Differential form.

• Fundamental theorem for line integrals (Multivariable gradient)
• Green’s theorem (Double integral)
• Острогра́дский’s divergence theorem (Flux and Divergence)
• Stokes’s theorem (Circulation and Curl)

The general structure of these generalizations¹ is as follows

• The left hand side involves some kind of derivative(s) of functions (single variable or multivariable)
• The right hand side involves the values of the original functions at the boundary of the domain of integration, and one less integral

The fundamental theorems may be used to generate alternative generalisations of Integration by parts.

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Footnotes

1. When stated in the format of the second part of the fundamental theorem. ↩