Matrix of Gradients: Notation Explained

[aaaa202]

There is one point in my book, where I am confused about the notation. In index notation the equation is:

da_i = a_j∇_j u_i

In matrix notation I would write this as:

da = (a⋅∇)u

where the term in the parenthis is just a scalar or if you will the unit matrix multiplied by a scalar.

But my book rewrites this as:

da = a ⋅ ∇u (1)

where the latter is a matrix of gradients with elements A_ij = ∇_ju_i

I don't understand this last rewriting. If you choose to use this matrix of gradients shouldn't it be:

da = (∇u)a

Or maybe I'm misinterpreting (1). Isn't a in this case a row vector and the matrix of displacement gradients has for example on the first row: ∇_xu_x,∇_yu_x ,∇_zu_x. I would like it to be transposed to make meaning of the above.

 

[RUber]

Just so I know we are speaking the same language:

There is one point in my book, where I am confused about the notation. In index notation the equation is:

da_i = a_j∇_j u_i

i.e. $\begin{pmatrix} da_1\\ da_2 \\ da_3 \end{pmatrix} =\begin{pmatrix} a_1 \frac{\partial u_1}{\partial x_1} &a_2 \frac{\partial u_1}{\partial x_2}& a_3 \frac{\partial u_1}{\partial x_3} \\ a_1 \frac{\partial u_1}{\partial x_1} &a_2 \frac{\partial u_2}{\partial x_2}& a_3 \frac{\partial u_2}{\partial x_3} \\ a_1 \frac{\partial u_3}{\partial x_1} &a_2 \frac{\partial u_3}{\partial x_2}& a_3 \frac{\partial u_3}{\partial x_3} \end{pmatrix}$

In matrix notation I would write this as:

da = (a⋅∇)u

where the term in the parenthis is just a scalar or if you will the unit matrix multiplied by a scalar.

How exactly would you define $(a \cdot \nabla)$ as a scalar?

But my book rewrites this as:

da = a ⋅ ∇u (1)

where the latter is a matrix of gradients with elements A_ij = ∇_ju_i

I don't understand this last rewriting. If you choose to use this matrix of gradients shouldn't it be:

da = (∇u)a

Or maybe I'm misinterpreting (1). Isn't a in this case a row vector and the matrix of displacement gradients has for example on the first row: ∇_xu_x,∇_yu_x ,∇_zu_x. I would like it to be transposed to make meaning of the above.

You are saying that $\nabla u = \begin{pmatrix} \frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2}& \frac{\partial u_1}{\partial x_3} \\ \frac{\partial u_1}{\partial x_1} & \frac{\partial u_2}{\partial x_2}& \frac{\partial u_2}{\partial x_3} \\ \frac{\partial u_3}{\partial x_1} & \frac{\partial u_3}{\partial x_2}& \frac{\partial u_3}{\partial x_3} \end{pmatrix}$
and $a = \begin{pmatrix} a_1 & a_2 & a_3 \end{pmatrix}$
So multiplying would have to be done by $a^T$, but that would give you a 3x1 matrix out, and appears to be equivalent to:
$\begin{pmatrix} da_1\\ da_2 \\ da_3 \end{pmatrix} = \begin{pmatrix} a_1 \frac{\partial u_1}{\partial x_1} + a_2 \frac{\partial u_1}{\partial x_2} + a_3 \frac{\partial u_1}{\partial x_3} \\ a_1 \frac{\partial u_1}{\partial x_1} + a_2 \frac{\partial u_2}{\partial x_2}+ a_3 \frac{\partial u_2}{\partial x_3} \\ a_1 \frac{\partial u_3}{\partial x_1} + a_2 \frac{\partial u_3}{\partial x_2}+ a_3 \frac{\partial u_3}{\partial x_3} \end{pmatrix}$

In general, matrix notation is flexible as long as you make sure your dimensions match with the operation you are trying to use. Most physics texts love to vector operations whereas many math and stats texts multiply by transpose matrices. The end result is the same.

 

[aaaa202]

I agree with the last result is what I want to get. But does that match with what you get in (1)? If I multiply the row vector a from the right with the matrix of gradients you have written, I don't get the last matrix in the above. Do you? Maybe I am simply failing to multiply the row vector by a matrix.
Also I would write the dot product as a scalar in index notation as a_j∇_j