# Product of a third- and second-order tensor.

• Nimlidor
In summary, this equation relates the vector of polarization (P) and stress tensor (T) with d being a third-order piezoelectric tensor.
Nimlidor
Hi,
I have read that vector of polarisation $\vec{P}$ and stress tensor T are linked with equation
$\vec{P}=\underline{\underline{d}} \: \underline{T}$
where d is a third-order piezoelectric tensor. Can anyone explain how you multiply a third and a second-order tensor to get a vector? A good link will also help.
Thank you,
N.

I'm not familiar with this equation, but I'm guessing that it's a contraction: ##A_{i}=\sum_{j,k}B_{ijk}C_{jk}##.

Also, it's usually a good idea to include a better reference than "I have read...". It might be easier to answer if we knew where you read it. If possible, you should link directly to the relevant page at Google Books.

So if I understand this correctly, in order to get "i-th" component of a vector you have to take i-th "slice" of third-order tensor and multiply it with second order tensor: not in the standard way we multiply matrices, but rather just multiply j,k-th component of both tensors and sum over all nine components (assuming that both tensors are three dimensional)?
I would have included a reference, but I only have it in printed version in Slovenian.

Nimlidor said:
So if I understand this correctly, in order to get "i-th" component of a vector you have to take i-th "slice" of third-order tensor and multiply it with second order tensor: not in the standard way we multiply matrices, but rather just multiply j,k-th component of both tensors and sum over all nine components (assuming that both tensors are three dimensional)?
You certainly don't have to do this to obtain the ith component of an arbitrary vector. What I'm saying is if you take a tensor with 3 indices and one with 2, and "contract" two indices as I did, the result is a component of a vector.

Does contracting over two indices mean that you make a sum of nine products?
$B_{i11}C_{11}+B_{i12}C_{12}+B_{i13}C_{13}+...+B_{i32}C_{32}+B_{i33}C_{33}$

Yes, that's what it means.

Thank you very much!

## 1. What is a third- and second-order tensor?

A third- and second-order tensor is a mathematical object that represents the relationships between vectors and scalars in a 3-dimensional space. It consists of multiple components that can be expressed as a matrix or an array of numbers.

## 2. How is the product of a third- and second-order tensor calculated?

The product of a third- and second-order tensor is calculated by multiplying the components of the two tensors according to a specific rule. This rule can vary depending on the context and type of tensors being multiplied.

## 3. What is the significance of the product of a third- and second-order tensor?

The product of a third- and second-order tensor is significant in various fields of science and engineering, such as mechanics, fluid dynamics, and electromagnetism. It allows for the representation and manipulation of complex relationships between multiple variables in a mathematical form.

## 4. Can the product of a third- and second-order tensor be visualized?

Yes, the product of a third- and second-order tensor can be visualized using graphical representations, such as diagrams or graphs. These visualizations can help in understanding the relationships between the components of the tensors and how they affect each other.

## 5. How is the product of a third- and second-order tensor used in real-world applications?

The product of a third- and second-order tensor is used in various real-world applications, such as analyzing stresses and strains in materials, predicting fluid flow in pipes, and modeling electromagnetic fields. It is an essential tool in solving complex problems in science and engineering.

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