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Date: 1-10-2019
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Date: 13-8-2018
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Date: 23-8-2019
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The first fundamental theorem of calculus states that, if is continuous on the closed interval
and
is the indefinite integral of
on
, then
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(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
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(2) |
then
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(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
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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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