How Do You Convert Complex Tensor Notation to Vector Notation?

[lostidentity]

Hi,

I have the following term in tensor notation

$$\frac{\partial{c}}{\partial{x_i}}\frac{\partial{u_i}}{\partial{x_j}}\frac{\partial{c}}{\partial{x_j}}$$

I'm not sure how to write this in vector notation.

Would it be?

$$\nabla{c}\cdot\nabla\boldsymbol{u}\cdot{c}$$

The problem I have is $$\nabla\boldsymbol{u}$$ is a tensor, whereas $$\nabla{c}$$ is a vector. Not sure what type of multiplication it would be between a vector and a tensor. Surely not a simple dot product?

Thanks.

 

[nicksauce]

This doesn't look like a proper tensor expression to me. If you have a repeated index, it should be repeated on "top" and on "bottom", whereas you have it repeated on the bottom both times here.

 

[lostidentity]

Hi,

Sorry I don't think I defined the problem correctly. $$c$$ is a scalar field and $$\vec{u}$$ is a vector field.

I checked with a paper and it seems that what I've got for the vector notation is correct. However, I'm having difficulty with the following term

$$\frac{\partial^2{c}}{\partial{x_k}\partial{x_i}}\frac{\partial^2{c}}{\partial{x_k}\partial{x_i}}$$ -----(2)

At first site I thought this could be the product of two Laplacian of a scalar field, however then I found that the correct form for the Laplacian in index notation is

$$\frac{\partial^2{c}}{\partial{x_i}\partial{x_i}}$$

So how would I write the above term (2) in vector notation?

Thanks.

 

[lostidentity]

Could $$\frac{\partial^2{c}}{\partial{x_k}\partial{x_i}}$$ be a second order tensor?

Since $$\frac{\partial{c}}{\partial{x_i}}$$ is the gradient of c (i.e. a vector), therefore $$\frac{\partial}{\partial{x_k}}\left(\frac{\partial{c}}{\partial{x_i}}\right)$$ would be the gradient of a vector field, i.e. a second order tensor?

If I were to write this is in tensor notation would it be

$$\nabla(\nabla{c})$$?

Thanks.

 

[yossell]

$$\frac{\partial{c}}{\partial{x_i}}\frac{\partial{u_i}}{\partial{x_j}}\frac{\partial{c}}{\partial{x_j}}$$

I think nicksauce is right - I don't know how to read this to make it come out as a tensor equation. If we're summing over repeated indices, then some of the indices are in the wrong position. If there's no summation, but it's meant for a particular i j, it's not a tensor equation. Perhaps you could give us the whole problem or tell us the summation convention you're using.

In rectangular components, $$\nabla$$ is $$\partial /\partial x_1 + \partial /\partial x_2 + \partial /\partial x_3$$

but the equations you're writing it looks like you're focussing on one component only - unless there's some kind of implicit summation over indices you've got in mind. Perhaps for now write out the summation explicitly so we know what you've got in mind.

 

[lostidentity]

Thanks for the reply.

Actually I'm working with a transport equation for a scalar variable N that has the following form (I've ignored a number of terms and constant coefficients as I don't think they are relevant).

$${u_j}\frac{\partial{N}}{\partial{x_j}} = \frac{\partial^2{c}}{\partial{x_k}\partial{x_i}}\frac{\partial^2{c}}{\partial{x_k}\partial{x_i}} - \frac{\partial{c}}{\partial{x_i}}\frac{\partial{u_ i}}{\partial{x_j}}\frac{\partial{c}}{\partial{x_j} }$$

So perhaps the summation is done for the j-th component since that's what's on the LHS?

I was wondering if I could write this whole equation in tensor notation rather than in the index notation as above. So far what I've got is something like$$\vec{u}\cdot\nabla{N} =[ \nabla(\nabla{c})]^2 - \nabla{c}\cdot\nabla\vec{u}\cdot\nabla{c}$$