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Explanation of some tensors in more detail
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I’m an engineering student and I was taught to use some kind of tensors to describe properties of materials, however, I have been trying to learn more about them in more detail for a couple of days, so excuse me if I’m mathematically wrong with something during the post.
I understand that if we have a multilinear function, we can associate it with a tensor object. If $V$ is a vectorial space, then
$$
f:underbraceV times cdots times V_text$n$ times rightarrow mathbbR quad textassociates with quad T : underbraceV times cdots times V_text$n$ times rightarrow mathbbR
$$
where we say that $T$ is a tensor $n$ times covariant.
With this, we could say for example that the dot product between two vectors is a tensor two times covariant. The problem for my understanding is when considering generic multilinear applications such that
$$
f : underbraceV times cdots times V_text$n$ times rightarrow underbraceV times cdots times V_text$m$ times
$$
Here it’s supposed that we could get a tensor as
$$
T : Vunderbrace^* times cdots times V^*_text$m$ times times underbraceV times cdots times V_text$n$ times rightarrow mathbbR
$$
where $V^*$ is the dual space (let’s say that is the same as $V$ but with row vectors instead of column ones), and then $T$ is said to be $m$ times contravariant and $n$ times covariant, or a tensor of type $(m,n)$.
In this scenario I can see easily that a matrix $A$ that transforms one vector into another, could be seen as a tensor of type $(1,1)$, because
$$
A : V rightarrow V Rightarrow quad textleads to quad T : V^* times V rightarrow mathbbR
$$
So my question is, knowing that column vectors could be seen as tensors of type $(0,1)$ and row vectors as those of type $(1,0)$, how can I achieve this result with a similar way of thinking to the one used for matrix $A$ or the dot product?
linear-algebra tensors applications
edited Jul 24 at 9:17
md2perpe
5,83511022
asked Jul 23 at 23:12
Jaime_mc2
1017
add a comment |Â
up vote
0
down vote
favorite
I’m an engineering student and I was taught to use some kind of tensors to describe properties of materials, however, I have been trying to learn more about them in more detail for a couple of days, so excuse me if I’m mathematically wrong with something during the post.
I understand that if we have a multilinear function, we can associate it with a tensor object. If $V$ is a vectorial space, then
$$
f:underbraceV times cdots times V_text$n$ times rightarrow mathbbR quad textassociates with quad T : underbraceV times cdots times V_text$n$ times rightarrow mathbbR
$$
where we say that $T$ is a tensor $n$ times covariant.
With this, we could say for example that the dot product between two vectors is a tensor two times covariant. The problem for my understanding is when considering generic multilinear applications such that
$$
f : underbraceV times cdots times V_text$n$ times rightarrow underbraceV times cdots times V_text$m$ times
$$
Here it’s supposed that we could get a tensor as
$$
T : Vunderbrace^* times cdots times V^*_text$m$ times times underbraceV times cdots times V_text$n$ times rightarrow mathbbR
$$
where $V^*$ is the dual space (let’s say that is the same as $V$ but with row vectors instead of column ones), and then $T$ is said to be $m$ times contravariant and $n$ times covariant, or a tensor of type $(m,n)$.
In this scenario I can see easily that a matrix $A$ that transforms one vector into another, could be seen as a tensor of type $(1,1)$, because
$$
A : V rightarrow V Rightarrow quad textleads to quad T : V^* times V rightarrow mathbbR
$$
So my question is, knowing that column vectors could be seen as tensors of type $(0,1)$ and row vectors as those of type $(1,0)$, how can I achieve this result with a similar way of thinking to the one used for matrix $A$ or the dot product?
linear-algebra tensors applications
edited Jul 24 at 9:17
md2perpe
5,83511022
asked Jul 23 at 23:12
Jaime_mc2
1017
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I’m an engineering student and I was taught to use some kind of tensors to describe properties of materials, however, I have been trying to learn more about them in more detail for a couple of days, so excuse me if I’m mathematically wrong with something during the post.
I understand that if we have a multilinear function, we can associate it with a tensor object. If $V$ is a vectorial space, then
$$
f:underbraceV times cdots times V_text$n$ times rightarrow mathbbR quad textassociates with quad T : underbraceV times cdots times V_text$n$ times rightarrow mathbbR
$$
where we say that $T$ is a tensor $n$ times covariant.
With this, we could say for example that the dot product between two vectors is a tensor two times covariant. The problem for my understanding is when considering generic multilinear applications such that
$$
f : underbraceV times cdots times V_text$n$ times rightarrow underbraceV times cdots times V_text$m$ times
$$
Here it’s supposed that we could get a tensor as
$$
T : Vunderbrace^* times cdots times V^*_text$m$ times times underbraceV times cdots times V_text$n$ times rightarrow mathbbR
$$
where $V^*$ is the dual space (let’s say that is the same as $V$ but with row vectors instead of column ones), and then $T$ is said to be $m$ times contravariant and $n$ times covariant, or a tensor of type $(m,n)$.
In this scenario I can see easily that a matrix $A$ that transforms one vector into another, could be seen as a tensor of type $(1,1)$, because
$$
A : V rightarrow V Rightarrow quad textleads to quad T : V^* times V rightarrow mathbbR
$$
So my question is, knowing that column vectors could be seen as tensors of type $(0,1)$ and row vectors as those of type $(1,0)$, how can I achieve this result with a similar way of thinking to the one used for matrix $A$ or the dot product?
linear-algebra tensors applications
edited Jul 24 at 9:17
md2perpe
5,83511022
asked Jul 23 at 23:12
Jaime_mc2
1017
I’m an engineering student and I was taught to use some kind of tensors to describe properties of materials, however, I have been trying to learn more about them in more detail for a couple of days, so excuse me if I’m mathematically wrong with something during the post.
I understand that if we have a multilinear function, we can associate it with a tensor object. If $V$ is a vectorial space, then
$$
f:underbraceV times cdots times V_text$n$ times rightarrow mathbbR quad textassociates with quad T : underbraceV times cdots times V_text$n$ times rightarrow mathbbR
$$
where we say that $T$ is a tensor $n$ times covariant.
With this, we could say for example that the dot product between two vectors is a tensor two times covariant. The problem for my understanding is when considering generic multilinear applications such that
$$
f : underbraceV times cdots times V_text$n$ times rightarrow underbraceV times cdots times V_text$m$ times
$$
Here it’s supposed that we could get a tensor as
$$
T : Vunderbrace^* times cdots times V^*_text$m$ times times underbraceV times cdots times V_text$n$ times rightarrow mathbbR
$$
where $V^*$ is the dual space (let’s say that is the same as $V$ but with row vectors instead of column ones), and then $T$ is said to be $m$ times contravariant and $n$ times covariant, or a tensor of type $(m,n)$.
In this scenario I can see easily that a matrix $A$ that transforms one vector into another, could be seen as a tensor of type $(1,1)$, because
$$
A : V rightarrow V Rightarrow quad textleads to quad T : V^* times V rightarrow mathbbR
$$
So my question is, knowing that column vectors could be seen as tensors of type $(0,1)$ and row vectors as those of type $(1,0)$, how can I achieve this result with a similar way of thinking to the one used for matrix $A$ or the dot product?
linear-algebra tensors applications
edited Jul 24 at 9:17
md2perpe
5,83511022
asked Jul 23 at 23:12
Jaime_mc2
1017
edited Jul 24 at 9:17
md2perpe
5,83511022
edited Jul 24 at 9:17
md2perpe
5,83511022
edited Jul 24 at 9:17
md2perpe
5,83511022
5,83511022
asked Jul 23 at 23:12
Jaime_mc2
1017
asked Jul 23 at 23:12
Jaime_mc2
1017
asked Jul 23 at 23:12
Jaime_mc2
1017
1017
add a comment |Â
add a comment |Â
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