# What kind of product denotes :

1. May 21, 2014

### jure

Hi,
I'm reading a book about fluid dynamics and I found some strange product between tensors. It's written like this: v=S:∇I , where S and I are matrices and v is a vector. Symbol : usually denotes Frobenius inner product. In this case we have a product of a matrix with a tensor of rank 3 and the result is vector - so it's not classical Frobenius product.
I would like to know what kind of product this is (especially useful would be index notation).
Thanks for help.

2. May 21, 2014

### pasmith

I would guess a double contraction, which would turn a rank 5 tensor into a vector:$$v_i = S_{ijk} \frac{\partial I_k}{\partial x_j}$$

3. May 21, 2014

### George Jones

Staff Emeritus
I initially guessed something like this too, but then thought that

might indicate that S only has two indices.

4. May 21, 2014

### George Jones

Staff Emeritus
Maybe something like $\left( S \cdot \nabla \right) \cdot I$ with component form

$$v_k = S_{ij} \frac{\partial I_{ik}}{\partial x_j}?$$

5. May 21, 2014

### jure

Yes, something like this. I and S only have 2 indices. How do you know over which indices to sum? Why wouldn't you sum over i and k instead of i and j?