Mixing
How to apply linear independence on the infinitesimals?
$begingroup$
Let $F(x,y,z): mathbb{R}^{3} rightarrow mathbb{R}$, and $dF = frac{partial F}{partial x}dx + frac{partial F}{partial y}dy + frac{partial F}{partial z}dz$.
If $x,y,z$ are linear independent, how to use the defination of the linear independent to proof $nexistsleft(a,b,cright) neq left(frac{partial F}{partial x},frac{partial F}{partial y},frac{partial F}{partial z}right): dF = adx+bdy+cdz$.
Linear independence is defined on vectors. $dx,dy,dz$ , which are infintesimals, are scalars. Is there any method for appling the defination of the linear independence on them?
linear-algebra derivatives
asked Jan 10 at 8:19
Iven CJ7Iven CJ7
52
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add a comment |
$begingroup$
Let $F(x,y,z): mathbb{R}^{3} rightarrow mathbb{R}$, and $dF = frac{partial F}{partial x}dx + frac{partial F}{partial y}dy + frac{partial F}{partial z}dz$.
If $x,y,z$ are linear independent, how to use the defination of the linear independent to proof $nexistsleft(a,b,cright) neq left(frac{partial F}{partial x},frac{partial F}{partial y},frac{partial F}{partial z}right): dF = adx+bdy+cdz$.
Linear independence is defined on vectors. $dx,dy,dz$ , which are infintesimals, are scalars. Is there any method for appling the defination of the linear independence on them?
linear-algebra derivatives
asked Jan 10 at 8:19
Iven CJ7Iven CJ7
52
$endgroup$
add a comment |
$begingroup$
Let $F(x,y,z): mathbb{R}^{3} rightarrow mathbb{R}$, and $dF = frac{partial F}{partial x}dx + frac{partial F}{partial y}dy + frac{partial F}{partial z}dz$.
If $x,y,z$ are linear independent, how to use the defination of the linear independent to proof $nexistsleft(a,b,cright) neq left(frac{partial F}{partial x},frac{partial F}{partial y},frac{partial F}{partial z}right): dF = adx+bdy+cdz$.
Linear independence is defined on vectors. $dx,dy,dz$ , which are infintesimals, are scalars. Is there any method for appling the defination of the linear independence on them?
linear-algebra derivatives
asked Jan 10 at 8:19
Iven CJ7Iven CJ7
52
$endgroup$
Let $F(x,y,z): mathbb{R}^{3} rightarrow mathbb{R}$, and $dF = frac{partial F}{partial x}dx + frac{partial F}{partial y}dy + frac{partial F}{partial z}dz$.
If $x,y,z$ are linear independent, how to use the defination of the linear independent to proof $nexistsleft(a,b,cright) neq left(frac{partial F}{partial x},frac{partial F}{partial y},frac{partial F}{partial z}right): dF = adx+bdy+cdz$.
Linear independence is defined on vectors. $dx,dy,dz$ , which are infintesimals, are scalars. Is there any method for appling the defination of the linear independence on them?
linear-algebra derivatives
linear-algebra derivatives
asked Jan 10 at 8:19
Iven CJ7Iven CJ7
52
asked Jan 10 at 8:19
Iven CJ7Iven CJ7
52
asked Jan 10 at 8:19
Iven CJ7Iven CJ7
52
asked Jan 10 at 8:19
Iven CJ7Iven CJ7
52
asked Jan 10 at 8:19
Iven CJ7Iven CJ7
52
52
add a comment |
add a comment |
1 Answer
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Actually, linear independence is defined in every vector space (or more general module). Here, you are looking at the tangent space which is spanned by the partial derivatives. Since they form a basis of the tangent space, you know that every vector in the tangent space, e.g. $dF$, admits a unique representation as linear combination in the basis. Consequently, $dx, dy, dz$ is the only possibility.
answered Jan 10 at 8:35
JamesJames
873215
$endgroup$
$begingroup$
I am not familiar with the tangent space. Are there any recommended books for studying?
$endgroup$
– Iven CJ7
Jan 10 at 16:22
$begingroup$
This isn’t really necessary for your purposes. You can just take as definition that it is the vector space spanned by the partial derivatives. But you will find more information in every basic book on differential geometry or algebraic geometry (i recommend looking at differential geometry first).
$endgroup$
– James
Jan 11 at 8:12
$begingroup$
I got it! Thanks a lot!
$endgroup$
– Iven CJ7
Jan 11 at 9:36
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Actually, linear independence is defined in every vector space (or more general module). Here, you are looking at the tangent space which is spanned by the partial derivatives. Since they form a basis of the tangent space, you know that every vector in the tangent space, e.g. $dF$, admits a unique representation as linear combination in the basis. Consequently, $dx, dy, dz$ is the only possibility.
answered Jan 10 at 8:35
JamesJames
873215
$endgroup$
$begingroup$
I am not familiar with the tangent space. Are there any recommended books for studying?
$endgroup$
– Iven CJ7
Jan 10 at 16:22
$begingroup$
This isn’t really necessary for your purposes. You can just take as definition that it is the vector space spanned by the partial derivatives. But you will find more information in every basic book on differential geometry or algebraic geometry (i recommend looking at differential geometry first).
$endgroup$
– James
Jan 11 at 8:12
$begingroup$
I got it! Thanks a lot!
$endgroup$
– Iven CJ7
Jan 11 at 9:36
add a comment |
$begingroup$
Actually, linear independence is defined in every vector space (or more general module). Here, you are looking at the tangent space which is spanned by the partial derivatives. Since they form a basis of the tangent space, you know that every vector in the tangent space, e.g. $dF$, admits a unique representation as linear combination in the basis. Consequently, $dx, dy, dz$ is the only possibility.
answered Jan 10 at 8:35
JamesJames
873215
$endgroup$
$begingroup$
I am not familiar with the tangent space. Are there any recommended books for studying?
$endgroup$
– Iven CJ7
Jan 10 at 16:22
$begingroup$
This isn’t really necessary for your purposes. You can just take as definition that it is the vector space spanned by the partial derivatives. But you will find more information in every basic book on differential geometry or algebraic geometry (i recommend looking at differential geometry first).
$endgroup$
– James
Jan 11 at 8:12
$begingroup$
I got it! Thanks a lot!
$endgroup$
– Iven CJ7
Jan 11 at 9:36
add a comment |
$begingroup$
Actually, linear independence is defined in every vector space (or more general module). Here, you are looking at the tangent space which is spanned by the partial derivatives. Since they form a basis of the tangent space, you know that every vector in the tangent space, e.g. $dF$, admits a unique representation as linear combination in the basis. Consequently, $dx, dy, dz$ is the only possibility.
answered Jan 10 at 8:35
JamesJames
873215
$endgroup$
Actually, linear independence is defined in every vector space (or more general module). Here, you are looking at the tangent space which is spanned by the partial derivatives. Since they form a basis of the tangent space, you know that every vector in the tangent space, e.g. $dF$, admits a unique representation as linear combination in the basis. Consequently, $dx, dy, dz$ is the only possibility.
answered Jan 10 at 8:35
JamesJames
873215
answered Jan 10 at 8:35
JamesJames
873215
answered Jan 10 at 8:35
JamesJames
873215
answered Jan 10 at 8:35
JamesJames
873215
873215
$begingroup$
I am not familiar with the tangent space. Are there any recommended books for studying?
$endgroup$
– Iven CJ7
Jan 10 at 16:22
$begingroup$
This isn’t really necessary for your purposes. You can just take as definition that it is the vector space spanned by the partial derivatives. But you will find more information in every basic book on differential geometry or algebraic geometry (i recommend looking at differential geometry first).
$endgroup$
– James
Jan 11 at 8:12
$begingroup$
I got it! Thanks a lot!
$endgroup$
– Iven CJ7
Jan 11 at 9:36
add a comment |
$begingroup$
I am not familiar with the tangent space. Are there any recommended books for studying?
$endgroup$
– Iven CJ7
Jan 10 at 16:22
$begingroup$
This isn’t really necessary for your purposes. You can just take as definition that it is the vector space spanned by the partial derivatives. But you will find more information in every basic book on differential geometry or algebraic geometry (i recommend looking at differential geometry first).
$endgroup$
– James
Jan 11 at 8:12
$begingroup$
I got it! Thanks a lot!
$endgroup$
– Iven CJ7
Jan 11 at 9:36
$begingroup$
I am not familiar with the tangent space. Are there any recommended books for studying?
$endgroup$
– Iven CJ7
Jan 10 at 16:22
$begingroup$
I am not familiar with the tangent space. Are there any recommended books for studying?
$endgroup$
– Iven CJ7
Jan 10 at 16:22
$begingroup$
This isn’t really necessary for your purposes. You can just take as definition that it is the vector space spanned by the partial derivatives. But you will find more information in every basic book on differential geometry or algebraic geometry (i recommend looking at differential geometry first).
$endgroup$
– James
Jan 11 at 8:12
$begingroup$
This isn’t really necessary for your purposes. You can just take as definition that it is the vector space spanned by the partial derivatives. But you will find more information in every basic book on differential geometry or algebraic geometry (i recommend looking at differential geometry first).
$endgroup$
– James
Jan 11 at 8:12
$begingroup$
I got it! Thanks a lot!
$endgroup$
– Iven CJ7
Jan 11 at 9:36
$begingroup$
I got it! Thanks a lot!
$endgroup$
– Iven CJ7
Jan 11 at 9:36
add a comment |
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