How to relate two definitions of space of 1-forms?
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I am trying to understand how to connect following quote from S. Carroll's "Spacetime and Geometry" with another definition of space of 1-forms from this book.
Roughly speaking, the space of one-forms at p is equivalent to the space of all functions that vanish at p and have the same second partial derivatives
Before that space of 1-forms was defined as the set of linear maps $omega: T_p to R$.
$$df(frac{partial}{partial lambda}) = frac{partial f}{partial lambda}$$
Linearity implies that map $omega$ preserves operation of addition and multiplication by number:
$$df(frac{partial}{partial lambda_1}+frac{partial}{partial lambda_2}) = df(frac{partial}{partial lambda_1})+df(frac{partial}{partial lambda_2})$$
$$df(acdot frac{partial}{partial lambda})= acdot df( frac{partial}{partial lambda})$$
My idea was to show that above definitions imply the statement of the quote. Yet I do not understand what is meant by "functions" in the quote.
At first I thought that they've meant $df$ itself. However it seems to be pointless that $df$ vanishes at p.
Then I thought that they were talking about function f.
I tried to rewrite $df$ in coordinate basis for simple case of 2d space:
$$df = frac{partial f}{partial x_1} dx_1 + frac{partial f}{partial x_2} dx_2$$
Still combining above definition with condition of linearity of 1-forms doesn't seem to give desired constrains on f.
I also find it fishy that f should have such severe constrains. I thought that $df$ could be defined for any (differentiable?) function on Manifold M.
Is the general direction of my reasoning correct? Or do I miss some essential points?
differential-forms general-relativity
asked Jan 23 at 19:24
Yaroslav ShustrovYaroslav Shustrov
158128
$endgroup$
add a comment |
$begingroup$
I am trying to understand how to connect following quote from S. Carroll's "Spacetime and Geometry" with another definition of space of 1-forms from this book.
Roughly speaking, the space of one-forms at p is equivalent to the space of all functions that vanish at p and have the same second partial derivatives
Before that space of 1-forms was defined as the set of linear maps $omega: T_p to R$.
$$df(frac{partial}{partial lambda}) = frac{partial f}{partial lambda}$$
Linearity implies that map $omega$ preserves operation of addition and multiplication by number:
$$df(frac{partial}{partial lambda_1}+frac{partial}{partial lambda_2}) = df(frac{partial}{partial lambda_1})+df(frac{partial}{partial lambda_2})$$
$$df(acdot frac{partial}{partial lambda})= acdot df( frac{partial}{partial lambda})$$
My idea was to show that above definitions imply the statement of the quote. Yet I do not understand what is meant by "functions" in the quote.
At first I thought that they've meant $df$ itself. However it seems to be pointless that $df$ vanishes at p.
Then I thought that they were talking about function f.
I tried to rewrite $df$ in coordinate basis for simple case of 2d space:
$$df = frac{partial f}{partial x_1} dx_1 + frac{partial f}{partial x_2} dx_2$$
Still combining above definition with condition of linearity of 1-forms doesn't seem to give desired constrains on f.
I also find it fishy that f should have such severe constrains. I thought that $df$ could be defined for any (differentiable?) function on Manifold M.
Is the general direction of my reasoning correct? Or do I miss some essential points?
differential-forms general-relativity
asked Jan 23 at 19:24
Yaroslav ShustrovYaroslav Shustrov
158128
$endgroup$
1
$begingroup$
Usually the starting point is for any $C^1$ curve $gamma : [0,1] to Vsubset M$ and functions $f,g : V to mathbb{R}$ let $f_gamma(t)=f(gamma(t))$ and define the $1$-form $g df$ by $int_gamma gdf = int_0^1 g_gamma(t) d f_gamma(t) = int_0^1 g_gamma(t) f_gamma'(t)dt$. The space of $1$-forms are the linear combination of those $g df$. The rules of integration tells the defining relations : $d(fg) = f dg+g df$, $d(f+g) = df+dg, (g+h)df = g df+hdf$, $dg=0$ iff $g$ is constant. For any chart $x : V to mathbb{R}^n$ any $1$-form is of the form $sum_{j=1}^n f_j d x_j$
$endgroup$
– reuns
Jan 23 at 20:11
add a comment |
$begingroup$
I am trying to understand how to connect following quote from S. Carroll's "Spacetime and Geometry" with another definition of space of 1-forms from this book.
Roughly speaking, the space of one-forms at p is equivalent to the space of all functions that vanish at p and have the same second partial derivatives
Before that space of 1-forms was defined as the set of linear maps $omega: T_p to R$.
$$df(frac{partial}{partial lambda}) = frac{partial f}{partial lambda}$$
Linearity implies that map $omega$ preserves operation of addition and multiplication by number:
$$df(frac{partial}{partial lambda_1}+frac{partial}{partial lambda_2}) = df(frac{partial}{partial lambda_1})+df(frac{partial}{partial lambda_2})$$
$$df(acdot frac{partial}{partial lambda})= acdot df( frac{partial}{partial lambda})$$
My idea was to show that above definitions imply the statement of the quote. Yet I do not understand what is meant by "functions" in the quote.
At first I thought that they've meant $df$ itself. However it seems to be pointless that $df$ vanishes at p.
Then I thought that they were talking about function f.
I tried to rewrite $df$ in coordinate basis for simple case of 2d space:
$$df = frac{partial f}{partial x_1} dx_1 + frac{partial f}{partial x_2} dx_2$$
Still combining above definition with condition of linearity of 1-forms doesn't seem to give desired constrains on f.
I also find it fishy that f should have such severe constrains. I thought that $df$ could be defined for any (differentiable?) function on Manifold M.
Is the general direction of my reasoning correct? Or do I miss some essential points?
differential-forms general-relativity
asked Jan 23 at 19:24
Yaroslav ShustrovYaroslav Shustrov
158128
$endgroup$
I am trying to understand how to connect following quote from S. Carroll's "Spacetime and Geometry" with another definition of space of 1-forms from this book.
Roughly speaking, the space of one-forms at p is equivalent to the space of all functions that vanish at p and have the same second partial derivatives
Before that space of 1-forms was defined as the set of linear maps $omega: T_p to R$.
$$df(frac{partial}{partial lambda}) = frac{partial f}{partial lambda}$$
Linearity implies that map $omega$ preserves operation of addition and multiplication by number:
$$df(frac{partial}{partial lambda_1}+frac{partial}{partial lambda_2}) = df(frac{partial}{partial lambda_1})+df(frac{partial}{partial lambda_2})$$
$$df(acdot frac{partial}{partial lambda})= acdot df( frac{partial}{partial lambda})$$
My idea was to show that above definitions imply the statement of the quote. Yet I do not understand what is meant by "functions" in the quote.
At first I thought that they've meant $df$ itself. However it seems to be pointless that $df$ vanishes at p.
Then I thought that they were talking about function f.
I tried to rewrite $df$ in coordinate basis for simple case of 2d space:
$$df = frac{partial f}{partial x_1} dx_1 + frac{partial f}{partial x_2} dx_2$$
Still combining above definition with condition of linearity of 1-forms doesn't seem to give desired constrains on f.
I also find it fishy that f should have such severe constrains. I thought that $df$ could be defined for any (differentiable?) function on Manifold M.
Is the general direction of my reasoning correct? Or do I miss some essential points?
differential-forms general-relativity
differential-forms general-relativity
asked Jan 23 at 19:24
Yaroslav ShustrovYaroslav Shustrov
158128
asked Jan 23 at 19:24
Yaroslav ShustrovYaroslav Shustrov
158128
asked Jan 23 at 19:24
Yaroslav ShustrovYaroslav Shustrov
158128
asked Jan 23 at 19:24
Yaroslav ShustrovYaroslav Shustrov
158128
asked Jan 23 at 19:24
Yaroslav ShustrovYaroslav Shustrov
158128
158128
1
$begingroup$
Usually the starting point is for any $C^1$ curve $gamma : [0,1] to Vsubset M$ and functions $f,g : V to mathbb{R}$ let $f_gamma(t)=f(gamma(t))$ and define the $1$-form $g df$ by $int_gamma gdf = int_0^1 g_gamma(t) d f_gamma(t) = int_0^1 g_gamma(t) f_gamma'(t)dt$. The space of $1$-forms are the linear combination of those $g df$. The rules of integration tells the defining relations : $d(fg) = f dg+g df$, $d(f+g) = df+dg, (g+h)df = g df+hdf$, $dg=0$ iff $g$ is constant. For any chart $x : V to mathbb{R}^n$ any $1$-form is of the form $sum_{j=1}^n f_j d x_j$
$endgroup$
– reuns
Jan 23 at 20:11
add a comment |
1
$begingroup$
Usually the starting point is for any $C^1$ curve $gamma : [0,1] to Vsubset M$ and functions $f,g : V to mathbb{R}$ let $f_gamma(t)=f(gamma(t))$ and define the $1$-form $g df$ by $int_gamma gdf = int_0^1 g_gamma(t) d f_gamma(t) = int_0^1 g_gamma(t) f_gamma'(t)dt$. The space of $1$-forms are the linear combination of those $g df$. The rules of integration tells the defining relations : $d(fg) = f dg+g df$, $d(f+g) = df+dg, (g+h)df = g df+hdf$, $dg=0$ iff $g$ is constant. For any chart $x : V to mathbb{R}^n$ any $1$-form is of the form $sum_{j=1}^n f_j d x_j$
$endgroup$
– reuns
Jan 23 at 20:11
1
1
$begingroup$
Usually the starting point is for any $C^1$ curve $gamma : [0,1] to Vsubset M$ and functions $f,g : V to mathbb{R}$ let $f_gamma(t)=f(gamma(t))$ and define the $1$-form $g df$ by $int_gamma gdf = int_0^1 g_gamma(t) d f_gamma(t) = int_0^1 g_gamma(t) f_gamma'(t)dt$. The space of $1$-forms are the linear combination of those $g df$. The rules of integration tells the defining relations : $d(fg) = f dg+g df$, $d(f+g) = df+dg, (g+h)df = g df+hdf$, $dg=0$ iff $g$ is constant. For any chart $x : V to mathbb{R}^n$ any $1$-form is of the form $sum_{j=1}^n f_j d x_j$
$endgroup$
– reuns
Jan 23 at 20:11
$begingroup$
Usually the starting point is for any $C^1$ curve $gamma : [0,1] to Vsubset M$ and functions $f,g : V to mathbb{R}$ let $f_gamma(t)=f(gamma(t))$ and define the $1$-form $g df$ by $int_gamma gdf = int_0^1 g_gamma(t) d f_gamma(t) = int_0^1 g_gamma(t) f_gamma'(t)dt$. The space of $1$-forms are the linear combination of those $g df$. The rules of integration tells the defining relations : $d(fg) = f dg+g df$, $d(f+g) = df+dg, (g+h)df = g df+hdf$, $dg=0$ iff $g$ is constant. For any chart $x : V to mathbb{R}^n$ any $1$-form is of the form $sum_{j=1}^n f_j d x_j$
$endgroup$
– reuns
Jan 23 at 20:11
add a comment |
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1
$begingroup$
Usually the starting point is for any $C^1$ curve $gamma : [0,1] to Vsubset M$ and functions $f,g : V to mathbb{R}$ let $f_gamma(t)=f(gamma(t))$ and define the $1$-form $g df$ by $int_gamma gdf = int_0^1 g_gamma(t) d f_gamma(t) = int_0^1 g_gamma(t) f_gamma'(t)dt$. The space of $1$-forms are the linear combination of those $g df$. The rules of integration tells the defining relations : $d(fg) = f dg+g df$, $d(f+g) = df+dg, (g+h)df = g df+hdf$, $dg=0$ iff $g$ is constant. For any chart $x : V to mathbb{R}^n$ any $1$-form is of the form $sum_{j=1}^n f_j d x_j$
$endgroup$
– reuns
Jan 23 at 20:11