Fundamental Theorems of Calculus
المؤلف:
Krantz, S. G.
المصدر:
"The Fundamental Theorem of Calculus along Curves." §2.1.5 in Handbook of Complex Variables. Boston, MA: Birkhäuser
الجزء والصفحة:
p. 22
20-8-2018
2299
Fundamental Theorems of Calculus
The first fundamental theorem of calculus states that, if 
is continuous on the closed interval ![[a,b]](http://mathworld.wolfram.com/images/equations/FundamentalTheoremsofCalculus/Inline2.gif)
and 
is the indefinite integral of 
on ![[a,b]](http://mathworld.wolfram.com/images/equations/FundamentalTheoremsofCalculus/Inline5.gif)
, then
 |
(1)
|
This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.
The second fundamental theorem of calculus holds for 
a continuous function on an open interval 
and 
any point in 
, and states that if 
is defined by
 |
(2)
|
then
 |
(3)
|
at each point in 
.
The fundamental theorem of calculus along curves states that if 
has a continuous indefinite integral 
in a region 
containing a parameterized curve 
for 
, then
REFERENCES:
Krantz, S. G. "The Fundamental Theorem of Calculus along Curves." §2.1.5 in Handbook of Complex Variables. Boston, MA: Birkhäuser, p. 22, 1999.
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