# Simple question or tensor notation

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
Homework Helper
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.

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.

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
Gold Member
$$\frac{\partial{c}}{\partial{x_i}}\frac{\partial{u_i}}{\partial{x_j}}\frac{\partial{c}}{\partial{x_j}}$$
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.

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

$$\vec{u}\cdot\nabla{N} =[ \nabla(\nabla{c})]^2 - \nabla{c}\cdot\nabla\vec{u}\cdot\nabla{c}$$