Clarification on Term Order of an Infinitesimal Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Determine the order of an infinitesimal.What is the order of an infinitesimal?Taylor series with functions as parameters (as opposed to variables)Calculus of Variations statement of a Singular Value Decomposition?What methods are available for this optimization problem?Derivative of an Infinitesimal?How should the Calculus of Variations deal with $delta(t-t_0)$ variations?Find the curve which together with $gamma$ encloses the greatest area.Where does Jacobi's accessory equation come from?Lie's theorem about infinitesimal rigid motions and riemannian manifolds curvatureUniqueness in Bernstein's theorem of calculus of variationsVariation of Lagrangian with infinitesimal transformation of time and position variables
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Clarification on Term Order of an Infinitesimal
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Determine the order of an infinitesimal.What is the order of an infinitesimal?Taylor series with functions as parameters (as opposed to variables)Calculus of Variations statement of a Singular Value Decomposition?What methods are available for this optimization problem?Derivative of an Infinitesimal?How should the Calculus of Variations deal with $delta(t-t_0)$ variations?Find the curve which together with $gamma$ encloses the greatest area.Where does Jacobi's accessory equation come from?Lie's theorem about infinitesimal rigid motions and riemannian manifolds curvatureUniqueness in Bernstein's theorem of calculus of variationsVariation of Lagrangian with infinitesimal transformation of time and position variables
$begingroup$
I've seen several questions posted on this, namely
What is the order of an infinitesimal?
Determine the order of an infinitesimal.
I'm asking this question because I've seen it pop up in variational calculus and want to make sure the answers in the above posts apply.
e.g. Gelfand & Fomin's Calculus of Variations states (Section 3.2, pg. 11), for
$$
beginalign
Delta J[h] &= J[y+h] - J[y] \
&=varphi [h] + epsilon lVert h rVert,
endalign
$$
that "the linear functional $varphi [h]$ which differs from $Delta J[h]$ by an infinitesimal of order higher than 1 relative to $lVert h rVert$, is called the [first] variation of $J[y]$ and is denoted by $delta J[h]$".
This kind of terminology appears more than once.
Question
Is the order of an infinitesimal, $epsilon$, its integer power, $n$, such that
$$
epsilon^n
$$
is an nth-order infinitesimal, with the only difference between orders of infinitesimals being how fast they converge to $0$?
(This definition being purely due to Cauchy, explained in his text, the English annotated version of which is given here:
http://users.uoa.gr/~spapast/TomeasDidaktikhs/Caychy/CauchyCoursdAnalyseAnAnnotatedTranslationSourcesandStudiesintheHistoryofMathematicsandPhysicalSciences.pdf))
calculus-of-variations
asked Mar 27 at 20:25
A. HendryA. Hendry
346
$endgroup$
add a comment |
$begingroup$
I've seen several questions posted on this, namely
What is the order of an infinitesimal?
Determine the order of an infinitesimal.
I'm asking this question because I've seen it pop up in variational calculus and want to make sure the answers in the above posts apply.
e.g. Gelfand & Fomin's Calculus of Variations states (Section 3.2, pg. 11), for
$$
beginalign
Delta J[h] &= J[y+h] - J[y] \
&=varphi [h] + epsilon lVert h rVert,
endalign
$$
that "the linear functional $varphi [h]$ which differs from $Delta J[h]$ by an infinitesimal of order higher than 1 relative to $lVert h rVert$, is called the [first] variation of $J[y]$ and is denoted by $delta J[h]$".
This kind of terminology appears more than once.
Question
Is the order of an infinitesimal, $epsilon$, its integer power, $n$, such that
$$
epsilon^n
$$
is an nth-order infinitesimal, with the only difference between orders of infinitesimals being how fast they converge to $0$?
(This definition being purely due to Cauchy, explained in his text, the English annotated version of which is given here:
http://users.uoa.gr/~spapast/TomeasDidaktikhs/Caychy/CauchyCoursdAnalyseAnAnnotatedTranslationSourcesandStudiesintheHistoryofMathematicsandPhysicalSciences.pdf))
calculus-of-variations
asked Mar 27 at 20:25
A. HendryA. Hendry
346
$endgroup$
add a comment |
$begingroup$
I've seen several questions posted on this, namely
What is the order of an infinitesimal?
Determine the order of an infinitesimal.
I'm asking this question because I've seen it pop up in variational calculus and want to make sure the answers in the above posts apply.
e.g. Gelfand & Fomin's Calculus of Variations states (Section 3.2, pg. 11), for
$$
beginalign
Delta J[h] &= J[y+h] - J[y] \
&=varphi [h] + epsilon lVert h rVert,
endalign
$$
that "the linear functional $varphi [h]$ which differs from $Delta J[h]$ by an infinitesimal of order higher than 1 relative to $lVert h rVert$, is called the [first] variation of $J[y]$ and is denoted by $delta J[h]$".
This kind of terminology appears more than once.
Question
Is the order of an infinitesimal, $epsilon$, its integer power, $n$, such that
$$
epsilon^n
$$
is an nth-order infinitesimal, with the only difference between orders of infinitesimals being how fast they converge to $0$?
(This definition being purely due to Cauchy, explained in his text, the English annotated version of which is given here:
http://users.uoa.gr/~spapast/TomeasDidaktikhs/Caychy/CauchyCoursdAnalyseAnAnnotatedTranslationSourcesandStudiesintheHistoryofMathematicsandPhysicalSciences.pdf))
calculus-of-variations
asked Mar 27 at 20:25
A. HendryA. Hendry
346
$endgroup$
I've seen several questions posted on this, namely
What is the order of an infinitesimal?
Determine the order of an infinitesimal.
I'm asking this question because I've seen it pop up in variational calculus and want to make sure the answers in the above posts apply.
e.g. Gelfand & Fomin's Calculus of Variations states (Section 3.2, pg. 11), for
$$
beginalign
Delta J[h] &= J[y+h] - J[y] \
&=varphi [h] + epsilon lVert h rVert,
endalign
$$
that "the linear functional $varphi [h]$ which differs from $Delta J[h]$ by an infinitesimal of order higher than 1 relative to $lVert h rVert$, is called the [first] variation of $J[y]$ and is denoted by $delta J[h]$".
This kind of terminology appears more than once.
Question
Is the order of an infinitesimal, $epsilon$, its integer power, $n$, such that
$$
epsilon^n
$$
is an nth-order infinitesimal, with the only difference between orders of infinitesimals being how fast they converge to $0$?
(This definition being purely due to Cauchy, explained in his text, the English annotated version of which is given here:
http://users.uoa.gr/~spapast/TomeasDidaktikhs/Caychy/CauchyCoursdAnalyseAnAnnotatedTranslationSourcesandStudiesintheHistoryofMathematicsandPhysicalSciences.pdf))
calculus-of-variations
calculus-of-variations
asked Mar 27 at 20:25
A. HendryA. Hendry
346
asked Mar 27 at 20:25
A. HendryA. Hendry
346
asked Mar 27 at 20:25
A. HendryA. Hendry
346
asked Mar 27 at 20:25
A. HendryA. Hendry
346
asked Mar 27 at 20:25
A. HendryA. Hendry
346
346
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes, the order of an infinitesimal is its integer power. The rate at which in converges to 0 is proportional to its order.
answered Apr 1 at 12:23
A. HendryA. Hendry
346
$endgroup$
add a comment |
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$begingroup$
Yes, the order of an infinitesimal is its integer power. The rate at which in converges to 0 is proportional to its order.
answered Apr 1 at 12:23
A. HendryA. Hendry
346
$endgroup$
add a comment |
$begingroup$
Yes, the order of an infinitesimal is its integer power. The rate at which in converges to 0 is proportional to its order.
answered Apr 1 at 12:23
A. HendryA. Hendry
346
$endgroup$
add a comment |
$begingroup$
Yes, the order of an infinitesimal is its integer power. The rate at which in converges to 0 is proportional to its order.
answered Apr 1 at 12:23
A. HendryA. Hendry
346
$endgroup$
Yes, the order of an infinitesimal is its integer power. The rate at which in converges to 0 is proportional to its order.
answered Apr 1 at 12:23
A. HendryA. Hendry
346
answered Apr 1 at 12:23
A. HendryA. Hendry
346
answered Apr 1 at 12:23
A. HendryA. Hendry
346
answered Apr 1 at 12:23
A. HendryA. Hendry
346
346
add a comment |
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