Prove that $frac{partial^{k} f}{partial x^{i_{1}} cdots partial x^{i_{k}}}$ is a tensor.
$begingroup$
Consider a function : $f(x^{1} dots x^{n}) ($$x^{i}- i$-th coordinate
$) in C^{infty}$ and $P$ such point : $forall 0<l <k$ $dfrac{partial ^{l} f}{partial x^{i_{1}} dots partial x^{i_{l}}}$ is zero at point $P$.
Now consider $A_{i_1 dots i_k} =dfrac{partial ^{k} f}{partial x^{i_{1}} dots partial x^{i_{k}}}$ we want to prove that it's a tensor. To prove this we need to check tensor-law:
$A_{i'_1 dots i'_k} =dfrac{partial x^{i_{1}}}{partial x^{i'_{1}}} dots dfrac{partial x^{i_{k}}}{partial x^{i'_{k}}} dfrac{partial ^{k} f}{partial x^{i_{1}} dots partial x^{i_{k}}}$.
I thought about considering $f = sum_{n_1 dots j_n} c(j_1 dots j_n) x_{i_1}^{j_1} dots x_{i_n}^{j_n}$ then condition about derivatives gives us some properties about coefficients.
Maybe there is a better way to prove it?
differential-geometry tensors
$endgroup$
|
show 1 more comment
$begingroup$
Consider a function : $f(x^{1} dots x^{n}) ($$x^{i}- i$-th coordinate
$) in C^{infty}$ and $P$ such point : $forall 0<l <k$ $dfrac{partial ^{l} f}{partial x^{i_{1}} dots partial x^{i_{l}}}$ is zero at point $P$.
Now consider $A_{i_1 dots i_k} =dfrac{partial ^{k} f}{partial x^{i_{1}} dots partial x^{i_{k}}}$ we want to prove that it's a tensor. To prove this we need to check tensor-law:
$A_{i'_1 dots i'_k} =dfrac{partial x^{i_{1}}}{partial x^{i'_{1}}} dots dfrac{partial x^{i_{k}}}{partial x^{i'_{k}}} dfrac{partial ^{k} f}{partial x^{i_{1}} dots partial x^{i_{k}}}$.
I thought about considering $f = sum_{n_1 dots j_n} c(j_1 dots j_n) x_{i_1}^{j_1} dots x_{i_n}^{j_n}$ then condition about derivatives gives us some properties about coefficients.
Maybe there is a better way to prove it?
differential-geometry tensors
$endgroup$
1
$begingroup$
Have you tried doing the case $k=2$ to see what's going on?
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:03
$begingroup$
@TedShifrin for case $k=2$ we have that all partial derivatives at point $P$ are zero. That's means that function behaves like : $dfrac{partial f}{partial x^{i}}$~$g(bar{x})(bar{x} -P)$
$endgroup$
– openspace
Dec 17 '18 at 19:13
$begingroup$
I would think more about how the second derivatives transform when you make a change of coordinates. I don't quite understand your statement.
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:33
$begingroup$
@TedShifrin you mean: $A_{i'_1, i'_2} = dfrac{partial x^{i_1}}{partial x^{i'_1}} dfrac{partial x^{i_2}}{partial x^{i'_2}} dfrac{partial^{2} f}{partial x^{i_1}partial x^{i_2}}$. That's part you are talking about ?
$endgroup$
– openspace
Dec 17 '18 at 20:30
$begingroup$
@TedShifrin the general question is to prove that $A_{i_1 dots i_k} = dfrac{partial^{k} f}{partial x^{i_1} dots partial x^{i_{k}}}$ is a tensor under conditions I've mentioned.
$endgroup$
– openspace
Dec 17 '18 at 20:41
|
show 1 more comment
$begingroup$
Consider a function : $f(x^{1} dots x^{n}) ($$x^{i}- i$-th coordinate
$) in C^{infty}$ and $P$ such point : $forall 0<l <k$ $dfrac{partial ^{l} f}{partial x^{i_{1}} dots partial x^{i_{l}}}$ is zero at point $P$.
Now consider $A_{i_1 dots i_k} =dfrac{partial ^{k} f}{partial x^{i_{1}} dots partial x^{i_{k}}}$ we want to prove that it's a tensor. To prove this we need to check tensor-law:
$A_{i'_1 dots i'_k} =dfrac{partial x^{i_{1}}}{partial x^{i'_{1}}} dots dfrac{partial x^{i_{k}}}{partial x^{i'_{k}}} dfrac{partial ^{k} f}{partial x^{i_{1}} dots partial x^{i_{k}}}$.
I thought about considering $f = sum_{n_1 dots j_n} c(j_1 dots j_n) x_{i_1}^{j_1} dots x_{i_n}^{j_n}$ then condition about derivatives gives us some properties about coefficients.
Maybe there is a better way to prove it?
differential-geometry tensors
$endgroup$
Consider a function : $f(x^{1} dots x^{n}) ($$x^{i}- i$-th coordinate
$) in C^{infty}$ and $P$ such point : $forall 0<l <k$ $dfrac{partial ^{l} f}{partial x^{i_{1}} dots partial x^{i_{l}}}$ is zero at point $P$.
Now consider $A_{i_1 dots i_k} =dfrac{partial ^{k} f}{partial x^{i_{1}} dots partial x^{i_{k}}}$ we want to prove that it's a tensor. To prove this we need to check tensor-law:
$A_{i'_1 dots i'_k} =dfrac{partial x^{i_{1}}}{partial x^{i'_{1}}} dots dfrac{partial x^{i_{k}}}{partial x^{i'_{k}}} dfrac{partial ^{k} f}{partial x^{i_{1}} dots partial x^{i_{k}}}$.
I thought about considering $f = sum_{n_1 dots j_n} c(j_1 dots j_n) x_{i_1}^{j_1} dots x_{i_n}^{j_n}$ then condition about derivatives gives us some properties about coefficients.
Maybe there is a better way to prove it?
differential-geometry tensors
differential-geometry tensors
edited Dec 17 '18 at 21:20
Lorenzo B.
1,8602520
1,8602520
asked Dec 17 '18 at 14:14
openspaceopenspace
3,4512822
3,4512822
1
$begingroup$
Have you tried doing the case $k=2$ to see what's going on?
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:03
$begingroup$
@TedShifrin for case $k=2$ we have that all partial derivatives at point $P$ are zero. That's means that function behaves like : $dfrac{partial f}{partial x^{i}}$~$g(bar{x})(bar{x} -P)$
$endgroup$
– openspace
Dec 17 '18 at 19:13
$begingroup$
I would think more about how the second derivatives transform when you make a change of coordinates. I don't quite understand your statement.
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:33
$begingroup$
@TedShifrin you mean: $A_{i'_1, i'_2} = dfrac{partial x^{i_1}}{partial x^{i'_1}} dfrac{partial x^{i_2}}{partial x^{i'_2}} dfrac{partial^{2} f}{partial x^{i_1}partial x^{i_2}}$. That's part you are talking about ?
$endgroup$
– openspace
Dec 17 '18 at 20:30
$begingroup$
@TedShifrin the general question is to prove that $A_{i_1 dots i_k} = dfrac{partial^{k} f}{partial x^{i_1} dots partial x^{i_{k}}}$ is a tensor under conditions I've mentioned.
$endgroup$
– openspace
Dec 17 '18 at 20:41
|
show 1 more comment
1
$begingroup$
Have you tried doing the case $k=2$ to see what's going on?
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:03
$begingroup$
@TedShifrin for case $k=2$ we have that all partial derivatives at point $P$ are zero. That's means that function behaves like : $dfrac{partial f}{partial x^{i}}$~$g(bar{x})(bar{x} -P)$
$endgroup$
– openspace
Dec 17 '18 at 19:13
$begingroup$
I would think more about how the second derivatives transform when you make a change of coordinates. I don't quite understand your statement.
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:33
$begingroup$
@TedShifrin you mean: $A_{i'_1, i'_2} = dfrac{partial x^{i_1}}{partial x^{i'_1}} dfrac{partial x^{i_2}}{partial x^{i'_2}} dfrac{partial^{2} f}{partial x^{i_1}partial x^{i_2}}$. That's part you are talking about ?
$endgroup$
– openspace
Dec 17 '18 at 20:30
$begingroup$
@TedShifrin the general question is to prove that $A_{i_1 dots i_k} = dfrac{partial^{k} f}{partial x^{i_1} dots partial x^{i_{k}}}$ is a tensor under conditions I've mentioned.
$endgroup$
– openspace
Dec 17 '18 at 20:41
1
1
$begingroup$
Have you tried doing the case $k=2$ to see what's going on?
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:03
$begingroup$
Have you tried doing the case $k=2$ to see what's going on?
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:03
$begingroup$
@TedShifrin for case $k=2$ we have that all partial derivatives at point $P$ are zero. That's means that function behaves like : $dfrac{partial f}{partial x^{i}}$~$g(bar{x})(bar{x} -P)$
$endgroup$
– openspace
Dec 17 '18 at 19:13
$begingroup$
@TedShifrin for case $k=2$ we have that all partial derivatives at point $P$ are zero. That's means that function behaves like : $dfrac{partial f}{partial x^{i}}$~$g(bar{x})(bar{x} -P)$
$endgroup$
– openspace
Dec 17 '18 at 19:13
$begingroup$
I would think more about how the second derivatives transform when you make a change of coordinates. I don't quite understand your statement.
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:33
$begingroup$
I would think more about how the second derivatives transform when you make a change of coordinates. I don't quite understand your statement.
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:33
$begingroup$
@TedShifrin you mean: $A_{i'_1, i'_2} = dfrac{partial x^{i_1}}{partial x^{i'_1}} dfrac{partial x^{i_2}}{partial x^{i'_2}} dfrac{partial^{2} f}{partial x^{i_1}partial x^{i_2}}$. That's part you are talking about ?
$endgroup$
– openspace
Dec 17 '18 at 20:30
$begingroup$
@TedShifrin you mean: $A_{i'_1, i'_2} = dfrac{partial x^{i_1}}{partial x^{i'_1}} dfrac{partial x^{i_2}}{partial x^{i'_2}} dfrac{partial^{2} f}{partial x^{i_1}partial x^{i_2}}$. That's part you are talking about ?
$endgroup$
– openspace
Dec 17 '18 at 20:30
$begingroup$
@TedShifrin the general question is to prove that $A_{i_1 dots i_k} = dfrac{partial^{k} f}{partial x^{i_1} dots partial x^{i_{k}}}$ is a tensor under conditions I've mentioned.
$endgroup$
– openspace
Dec 17 '18 at 20:41
$begingroup$
@TedShifrin the general question is to prove that $A_{i_1 dots i_k} = dfrac{partial^{k} f}{partial x^{i_1} dots partial x^{i_{k}}}$ is a tensor under conditions I've mentioned.
$endgroup$
– openspace
Dec 17 '18 at 20:41
|
show 1 more comment
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1
$begingroup$
Have you tried doing the case $k=2$ to see what's going on?
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:03
$begingroup$
@TedShifrin for case $k=2$ we have that all partial derivatives at point $P$ are zero. That's means that function behaves like : $dfrac{partial f}{partial x^{i}}$~$g(bar{x})(bar{x} -P)$
$endgroup$
– openspace
Dec 17 '18 at 19:13
$begingroup$
I would think more about how the second derivatives transform when you make a change of coordinates. I don't quite understand your statement.
$endgroup$
– Ted Shifrin
Dec 17 '18 at 19:33
$begingroup$
@TedShifrin you mean: $A_{i'_1, i'_2} = dfrac{partial x^{i_1}}{partial x^{i'_1}} dfrac{partial x^{i_2}}{partial x^{i'_2}} dfrac{partial^{2} f}{partial x^{i_1}partial x^{i_2}}$. That's part you are talking about ?
$endgroup$
– openspace
Dec 17 '18 at 20:30
$begingroup$
@TedShifrin the general question is to prove that $A_{i_1 dots i_k} = dfrac{partial^{k} f}{partial x^{i_1} dots partial x^{i_{k}}}$ is a tensor under conditions I've mentioned.
$endgroup$
– openspace
Dec 17 '18 at 20:41