A Natural Approach to the Fundamental Theorem of the Integral Calculus

1954 ◽  
Vol 61 (6) ◽  
pp. 413 ◽  
Author(s):  
J. P. Hoyt
Keyword(s):  
Fundamental Theorem ◽  
Integral Calculus ◽  
Natural Approach

scholarly journals Fundamental theorem of Wiener calculus

1990 ◽  
Vol 13 (3) ◽  
pp. 443-452
Author(s):  
Chull Park ◽  
David Skoug ◽  
Lawrence Smolowitz
Keyword(s):  
Fundamental Theorem ◽  
Wiener Space ◽  
Wiener Integral ◽  
Integral Calculus ◽  
Indefinite Integral

In this paper we define and develop a theory of differentiation in Wiener spaceC[0,T]. We then proceed to establish a fundamental theorem of the integral calculus forC[0,T]. First of all, we show that the derivative of the indefinite Wiener integral exists and equals the integrand functional. Secondly, we show that certain functionals defined onC[0,T]are equal to the indefinite integral of their Wiener derivative.

scholarly journals Fractional Line Integral

Mathematics ◽  
2021 ◽  
Vol 9 (10) ◽  
pp. 1150
Author(s):  
Gabriel Bengochea ◽  
Manuel Ortigueira
Keyword(s):  
Fundamental Theorem ◽  
Piecewise Linear ◽  
Directional Derivatives ◽  
Definite Integral ◽  
Integral Calculus ◽  
Regular Curve ◽  
Line Integral ◽  
Linear Path ◽  
Definition Of

This paper proposed a definition of the fractional line integral, generalising the concept of the fractional definite integral. The proposal replicated the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It was based on the concept of the fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integral, the Grünwald–Letnikov and Liouville directional derivatives were introduced and their properties described. The integral was defined for a piecewise linear path first and, from it, for any regular curve.

scholarly journals Generalized Nabla Differentiability and Integrability for Fuzzy Functions on Time Scales

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 65 ◽  
Author(s):  
R. Leelavathi ◽  
G. Suresh Kumar ◽  
Ravi P. Agarwal ◽  
Chao Wang ◽  
M.S.N. Murty
Keyword(s):  
Time Scales ◽  
Fundamental Theorem ◽  
Integral Calculus ◽  
Hukuhara Difference

This paper mainly deals with introducing and studying the properties of generalized nabla differentiability for fuzzy functions on time scales via Hukuhara difference. Further, we obtain embedding results on E n for generalized nabla differentiable fuzzy functions. Finally, we prove a fundamental theorem of a nabla integral calculus for fuzzy functions on time scales under generalized nabla differentiability. The obtained results are illustrated with suitable examples.

scholarly journals Integrability and the Integral of Partial Functions from R into R1

2006 ◽  
Vol 14 (4) ◽  
pp. 207-212
Author(s):  
Noboru Endou ◽  
Yasunari Shidama ◽  
Masahiko Yamazaki
Keyword(s):  
General Expression ◽  
Fundamental Theorem ◽  
Partial Functions ◽  
Integral Calculus ◽  
Fundamental Theorem Of Calculus ◽  
Indefinite Integral

Integrability and the Integral of Partial Functions from R into R1 In this paper, we showed the linearity of the indefinite integral the form of which was introduced in [11]. In addition, we proved some theorems about the integral calculus on the subinterval of [a,b]. As a result, we described the fundamental theorem of calculus, that we developed in [11], by a more general expression.

The fundamental theorem of the integral calculus

2011 ◽  
pp. 119-127
Author(s):  
Luigi Ambrosio ◽  
Giuseppe Da Prato ◽  
Andrea Mennucci
Keyword(s):  
Fundamental Theorem ◽  
Integral Calculus
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