Mixing
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Taylor series on manifolds
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Any analytic function $f: mathbb{R}tomathbb{R}$ can be written as the Taylor series:
$$f(x) = sum_{i=0}^inftyfrac{f^{(i)}(0)}{i!}x^i$$
I want to generalize this on manifolds.
However, if $f:Mto mathbb{R}$ is an analytic function, the derivatives at $p in M$ are
$$T_pf:TM to Tmathbb{R}$$
$$TT_pf:TTMto TTmathbb{R}$$
$$dots$$
, so serieses like the following are meaningless:
$$f(p)+frac{T_pf}{1}X+frac{T^2_pf}{2}X^2+dots$$
Is there a generalization of Taylor's series to manifolds?
calculus geometry manifolds taylor-expansion
asked Jan 27 at 17:20
satoukibisatoukibi
16718
$endgroup$
add a comment |
$begingroup$
Any analytic function $f: mathbb{R}tomathbb{R}$ can be written as the Taylor series:
$$f(x) = sum_{i=0}^inftyfrac{f^{(i)}(0)}{i!}x^i$$
I want to generalize this on manifolds.
However, if $f:Mto mathbb{R}$ is an analytic function, the derivatives at $p in M$ are
$$T_pf:TM to Tmathbb{R}$$
$$TT_pf:TTMto TTmathbb{R}$$
$$dots$$
, so serieses like the following are meaningless:
$$f(p)+frac{T_pf}{1}X+frac{T^2_pf}{2}X^2+dots$$
Is there a generalization of Taylor's series to manifolds?
calculus geometry manifolds taylor-expansion
asked Jan 27 at 17:20
satoukibisatoukibi
16718
$endgroup$
$begingroup$
Is there a good reason you don't just want to do this in charts?
$endgroup$
– user98602
Jan 27 at 18:56
$begingroup$
@MikeMiller I like coordinate-free expressions.
$endgroup$
– satoukibi
Jan 28 at 14:22
$begingroup$
I do too, but since you have to center at a point (and even for analytic functions on $Bbb R$, the Taylor series does not necessarily converge globally), it is hard for me to inagine this as anything but an essentially local theorem.
$endgroup$
– user98602
Jan 28 at 14:29
$begingroup$
I get it, it's an essentially local theorem.
$endgroup$
– satoukibi
Jan 28 at 14:56
add a comment |
$begingroup$
Any analytic function $f: mathbb{R}tomathbb{R}$ can be written as the Taylor series:
$$f(x) = sum_{i=0}^inftyfrac{f^{(i)}(0)}{i!}x^i$$
I want to generalize this on manifolds.
However, if $f:Mto mathbb{R}$ is an analytic function, the derivatives at $p in M$ are
$$T_pf:TM to Tmathbb{R}$$
$$TT_pf:TTMto TTmathbb{R}$$
$$dots$$
, so serieses like the following are meaningless:
$$f(p)+frac{T_pf}{1}X+frac{T^2_pf}{2}X^2+dots$$
Is there a generalization of Taylor's series to manifolds?
calculus geometry manifolds taylor-expansion
asked Jan 27 at 17:20
satoukibisatoukibi
16718
$endgroup$
Any analytic function $f: mathbb{R}tomathbb{R}$ can be written as the Taylor series:
$$f(x) = sum_{i=0}^inftyfrac{f^{(i)}(0)}{i!}x^i$$
I want to generalize this on manifolds.
However, if $f:Mto mathbb{R}$ is an analytic function, the derivatives at $p in M$ are
$$T_pf:TM to Tmathbb{R}$$
$$TT_pf:TTMto TTmathbb{R}$$
$$dots$$
, so serieses like the following are meaningless:
$$f(p)+frac{T_pf}{1}X+frac{T^2_pf}{2}X^2+dots$$
Is there a generalization of Taylor's series to manifolds?
calculus geometry manifolds taylor-expansion
calculus geometry manifolds taylor-expansion
asked Jan 27 at 17:20
satoukibisatoukibi
16718
asked Jan 27 at 17:20
satoukibisatoukibi
16718
asked Jan 27 at 17:20
satoukibisatoukibi
16718
asked Jan 27 at 17:20
satoukibisatoukibi
16718
asked Jan 27 at 17:20
satoukibisatoukibi
16718
16718
$begingroup$
Is there a good reason you don't just want to do this in charts?
$endgroup$
– user98602
Jan 27 at 18:56
$begingroup$
@MikeMiller I like coordinate-free expressions.
$endgroup$
– satoukibi
Jan 28 at 14:22
$begingroup$
I do too, but since you have to center at a point (and even for analytic functions on $Bbb R$, the Taylor series does not necessarily converge globally), it is hard for me to inagine this as anything but an essentially local theorem.
$endgroup$
– user98602
Jan 28 at 14:29
$begingroup$
I get it, it's an essentially local theorem.
$endgroup$
– satoukibi
Jan 28 at 14:56
add a comment |
$begingroup$
Is there a good reason you don't just want to do this in charts?
$endgroup$
– user98602
Jan 27 at 18:56
$begingroup$
@MikeMiller I like coordinate-free expressions.
$endgroup$
– satoukibi
Jan 28 at 14:22
$begingroup$
I do too, but since you have to center at a point (and even for analytic functions on $Bbb R$, the Taylor series does not necessarily converge globally), it is hard for me to inagine this as anything but an essentially local theorem.
$endgroup$
– user98602
Jan 28 at 14:29
$begingroup$
I get it, it's an essentially local theorem.
$endgroup$
– satoukibi
Jan 28 at 14:56
$begingroup$
Is there a good reason you don't just want to do this in charts?
$endgroup$
– user98602
Jan 27 at 18:56
$begingroup$
Is there a good reason you don't just want to do this in charts?
$endgroup$
– user98602
Jan 27 at 18:56
$begingroup$
@MikeMiller I like coordinate-free expressions.
$endgroup$
– satoukibi
Jan 28 at 14:22
$begingroup$
@MikeMiller I like coordinate-free expressions.
$endgroup$
– satoukibi
Jan 28 at 14:22
$begingroup$
I do too, but since you have to center at a point (and even for analytic functions on $Bbb R$, the Taylor series does not necessarily converge globally), it is hard for me to inagine this as anything but an essentially local theorem.
$endgroup$
– user98602
Jan 28 at 14:29
$begingroup$
I do too, but since you have to center at a point (and even for analytic functions on $Bbb R$, the Taylor series does not necessarily converge globally), it is hard for me to inagine this as anything but an essentially local theorem.
$endgroup$
– user98602
Jan 28 at 14:29
$begingroup$
I get it, it's an essentially local theorem.
$endgroup$
– satoukibi
Jan 28 at 14:56
$begingroup$
I get it, it's an essentially local theorem.
$endgroup$
– satoukibi
Jan 28 at 14:56
add a comment |
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$begingroup$
Is there a good reason you don't just want to do this in charts?
$endgroup$
– user98602
Jan 27 at 18:56
$begingroup$
@MikeMiller I like coordinate-free expressions.
$endgroup$
– satoukibi
Jan 28 at 14:22
$begingroup$
I do too, but since you have to center at a point (and even for analytic functions on $Bbb R$, the Taylor series does not necessarily converge globally), it is hard for me to inagine this as anything but an essentially local theorem.
$endgroup$
– user98602
Jan 28 at 14:29
$begingroup$
I get it, it's an essentially local theorem.
$endgroup$
– satoukibi
Jan 28 at 14:56