Tensor Product

Today, we're going to delve into tensor products. Can anyone tell me what they think happens when we combine two vectors?

Do we create another vector?

Good guess! However, when we combine vectors using the tensor product, we actually produce what we call a second order tensor. This result is different from vector products, which yield scalars or vectors.

So, tensors are more complex than just vectors?

Exactly! To represent the tensor product of two vectors $a \otimes b = C$, we can use a matrix form where each element is formed by multiplying the components of these vectors. For example, if $a = [a_1, a_2]$ and $b = [b_1, b_2]$, how would you calculate the elements of tensor C?

Would it be $C_{ij} = a_i b_j$?

Exactly! That’s the key formula for the tensor product. Remember, this operation leads to a second order tensor because we are producing a matrix.

What about the tensor's properties compared to the original vectors?

Great question! The properties of the tensor, including its dimensionality, are independent of the coordinate systems. In other words, while the representation of the tensor changes when we switch coordinates, the underlying tensor itself does not.

To summarize, we learned that the tensor product generates a second order tensor from two vectors, and its properties remain constant irrespective of coordinate transformations.

Now that we understand what a tensor product is, let's look at how we can operate with tensors. Can anyone explain what happens when we multiply a second order tensor with a vector?

Does it give us another vector?

Correct! Multiplying a second order tensor by a vector will yield a vector, where the multiplication is defined through the dot product of the tensor's second vector with the vector we are multiplying. Can anyone provide me with how this operation looks mathematically?

I think it’s something like $a = C b$?

Nice! So, if $C$ is represented in matrix form and $b$ is our vector, we’re essentially performing matrix-vector multiplication. Let’s illustrate this with an example. What would be your first step?

We would set up the matrix for tensor C and then multiply it by vector b?

Exactly! You can view this entire operation in the context of applying forces in mechanical systems represented by tensors. It simplifies complex calculations.

To recap, we discussed the interaction between tensors and vectors through multiplication, concluding that we indeed generate a new vector. This process highlights the critical role tensors play in many physical applications.

Let’s take it a step further and explore higher order tensors. What do you think a third order tensor might represent in our context?

Would it add another dimension of interaction between multiple vectors?

Exactly! A third order tensor would allow us to encode relationships across multiple dimensions. Similar to second order tensors, these can also be transformed across coordinate systems. Can anyone recall how we can express a general tensor's matrix representation?

Each tensor can be expressed as a linear combination of its basis tensors, right?

Spot on! In the case of a second order tensor, we deal with nine basis tensors. How does that relate to their representation in various coordinate systems?

The coefficients change based on how the basis tensors interact with the coordinate systems?

Yes! Each tensor component is independent, providing great flexibility for our calculations in complex systems. To summarize, we explored how a tensor can expand into higher orders, enriching its representation and capabilities in multidimensional problems.