A Noether's theorem time invariance -- mean value theorem use?

AI Thread Summary
The discussion revolves around the application of the mean value theorem (MVT) in the context of Noether's theorem and time invariance. Participants clarify that the MVT states that for a continuous function, the integral over an interval can be expressed as the product of the interval's length and the function's value at some point within that interval. There is some confusion regarding the correct formulation of the MVT, with references to various sources to validate the statements. The conversation emphasizes the relationship between the MVT for definite integrals and the general mean value theorem. Overall, the thread seeks to clarify how the MVT is utilized in the initial steps of applying Noether's theorem.
Tush19
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how does the first step use mean value theorem? I don't get it , can anyone explain , thanks.
 
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The mean value theorem states that
$$
\int_x^{x+\delta x} f(s) ds = \delta x\, f(x^*)
$$
where ##x \leq x^* \leq x + \delta x##. Since ##f## is continuous, ##f(x^*) \to f(x)## for small ##\delta x##.
 
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Orodruin said:
The mean value theorem states that
$$
\int_x^{x+\delta x} f(s) ds = \delta x\, f(x^*)
$$
where ##x \leq x^* \leq x + \delta x##. Since ##f## is continuous, ##f(x^*) \to f(x)## for small ##\delta x##.
thanks but I couldn't find that mean value theorem statement anywhere ,all it shows that mean value theorem is the following
1641479493425-png.png
 
Just to add: The mean value theorem for definite integrals is easy to obtain from the theorem you quoted. Just consider that
$$
(b-a) f’(c) = f(b) - f(a) = \int_a^b f’(x) dx
$$
and let ##g(x) = f’(x)##. You now have
$$
\int_a^b g(x) dx = (b-a) g(c)
$$
for some ##c## such that ##a\leq c\leq b##.
 
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Orodruin said:
Just to add: The mean value theorem for definite integrals is easy to obtain from the theorem you quoted. Just consider that
$$
(b-a) f’(c) = f(b) - f(a) = \int_a^b f’(x) dx
$$
and let ##g(x) = f’(x)##. You now have
$$
\int_a^b g(x) dx = (b-a) g(c)
$$
for some ##c## such that ##a\leq c\leq b##.
thank you so much
 

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