A Need some clarifications on tensor calculus please

pitaly
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Need some clarifications on tensor calculus
I've started reading up on tensors. Since this lies well outside my usual area, I need some clarifications on some tensor calculus issues.

Let ##A## be a tensor of order ##j > 1##. Suppose that the tensor is cubical, i.e., every mode is of the same size. So for example, if ##A## is of order 3 then ##A \in R^{n \times n \times n}##.

Further assume that ##A## is symmetric (sometimes called supersymmetric), i.e., it is invariant under permutation of indices. For example, if the order is 3 then ##A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kij}##.

We say that a tensor is positive semidefinite if ##x \cdot A \cdot x \geq 0## for all vectors ##x \neq 0##. So for example, if ##A## is of order 2, we have a standard quadratic form and if ##A## is positive semidefinite then ##x \cdot A \cdot x \geq 0##.

Now, to my two questions:

1) Is there any clean notation for tensor differentiation? If ##A## is of order 2 then the gradient of ##x \cdot A \cdot x## with respect to ##x## is ##2 x \cdot A##. How do I write the gradient of ##x \cdot A \cdot x## with respect to ##x## when ##A## is a tensor of any order ##j>2##?

2) Consider a tensor ##A## of order 2 and the quadratic form ##(x-y) \cdot A \cdot (x-y)## for any vectors ##x,y##. If ##A## is positive semidefinite then ##x \cdot A \cdot x - y \cdot A \cdot y \geq 2y \cdot A \cdot (x-y) ##. Does there exist any analogous inequality for tensors of any order ##j>2##, where the tensor ##A## is positive semidefinite?

Any help on this is highly appreciated!
 
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pitaly said:
Summary:: Need some clarifications on tensor calculus

I've started reading up on tensors. Since this lies well outside my usual area, I need some clarifications on some tensor calculus issues.

Let ##A## be a tensor of order ##j > 1##. Suppose that the tensor is cubical, i.e., every mode is of the same size. So for example, if ##A## is of order 3 then ##A \in R^{n \times n \times n}##.

Further assume that ##A## is symmetric (sometimes called supersymmetric), i.e., it is invariant under permutation of indices. For example, if the order is 3 then ##A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kij}##.

We say that a tensor is positive semidefinite if ##x \cdot A \cdot x \geq 0## for all vectors ##x \neq 0##. So for example, if ##A## is of order 2, we have a standard quadratic form and if ##A## is positive semidefinite then ##x \cdot A \cdot x \geq 0##.
What is ##x \cdot u \otimes v \otimes w \cdot x\;##?
pitaly said:
Now, to my two questions:

1) Is there any clean notation for tensor differentiation? If ##A## is of order 2 then the gradient of ##x \cdot A \cdot x## with respect to ##x## is ##2 x \cdot A##. How do I write the gradient of ##x \cdot A \cdot x## with respect to ##x## when ##A## is a tensor of any order ##j>2##?
The first that comes to my mind is the boundary operator in cohomology theory.
pitaly said:
2) Consider a tensor ##A## of order 2 and the quadratic form ##(x-y) \cdot A \cdot (x-y)## for any vectors ##x,y##. If ##A## is positive semidefinite then ##x \cdot A \cdot x - y \cdot A \cdot y \geq 2y \cdot A \cdot (x-y) ##. Does there exist any analogous inequality for tensors of any order ##j>2##, where the tensor ##A## is positive semidefinite?

Any help on this is highly appreciated!
Good question. I had the same (see above).

Let's see what you have actually done. You fed a tensor with two vectors and computed a scalar. Hence we need ##u^*,v^*\in V^*## to get ##(u^*\otimes v^*)(x,y)= u^*(x)v^*(y)\in \mathbb{R}.## We certainly can fill up this equation with arbitrary many ##w^{(k)}\in V## to get
$$
(u^*\otimes v^*\otimes w^{(1)}\otimes \ldots\otimes w^{(m)})(x,y)= u^*(x)v^*(y) \cdot w^{(1)}\otimes \ldots\otimes w^{(m)}
$$
but then we lose the possibility to compare it with the scalar ##0.##

I think you are approaching this from the wrong side. What do you want to achieve is the question you should ask, not how can I generalize something.

Since you asked in a mathematics forum, you may want to have a read:
https://www.physicsforums.com/insights/what-is-a-tensor/
https://www.physicsforums.com/insights/pantheon-derivatives-part-iii/
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/
 
pitaly said:
Summary:: Need some clarifications on tensor calculus

1) Is there any clean notation for tensor differentiation? If A is of order 2 then the gradient of x⋅A⋅x with respect to x is 2x⋅A. How do I write the gradient of x⋅A⋅x with respect to x when A is a tensor of any order j>2?
Not an expert, but I know my way around some basic tensor math. Normally I would the tensor product in Einstein notation, ie ##A_{ijk}x^ix^j##. Then I would write the derivative as a tensor-like expression: ##\partial_i = \frac{\partial}{\partial x^i}##. And then we can apply it:
$$\partial_l A_{ijk}x^ix^j = A_{ijk}\partial_l(x^ix^j)$$ Knowing the Leibniz rule and that ##\frac{\partial x^i}{\partial x^j}=\delta^i_j##, we can expand out the derivative: $$ \begin{align} \partial_l A_{ijk}x^ix^j & = A_{ijk}((\partial_l x^i)x^j + x^i\partial_l(x^j)) \\ & = A_{ijk}(\delta_l^ix^j+x^i\delta_l^j) \\ & = A_{ljk}x^j+A_{ilk}x^i\end{align}$$ In the case that ##A## is symmetric, we can freely swap the first two indices to get the expression ##A_{jlk}x^j+A_{ilk}x^i##, which are really two sums with the same form, so we can change the index to be the same:
$$A_{ilk}x^i+A_{ilk}x^i=2A_{ilk}x^i$$ I guess it looks kinda hand-wavey, but it’s really just a compact way of writing sums along indices. Add as many indices beyond ##l## to get the same identity for order >3.
 
pitaly said:
Further assume that ##A## is symmetric (sometimes called supersymmetric), i.e., it is invariant under permutation of indices. For example, if the order is 3 then ##A_{ijk} = A_{ikj} = A_{jik} = A_{jki} = A_{kij} = A_{kij}##.
I've never seen the expression "supersymmetric" for this, and seems very confusing; supersymmetry is something completely different. But maybe it's common in engineering texts.
 
pitaly said:
Summary:: Need some clarifications on tensor calculus

For example, if the order is 3 then Aijk=Aikj=Ajik=Ajki=Akij=Akij.
I think the last A term should be k j i .
 
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