Anybody know Einstein notation for divergence and curl?

What I would like to do is give each of these formulas in three forms, and then ask a fairly simple question; What is the Einstein notation for each of these formulas?

The unit vectors, in matrix notation:

[tex]\vec{u_x}=\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} or \begin{pmatrix} 1& 0& 0 \end{pmatrix}[/tex]

Similarly

[tex]\vec{u_y}=(0,1,0) , \vec{u_z}=(0,0,1)[/tex]

The "del" operator

[tex]\nabla = \vec{u_x} \frac{\partial }{\partial x} + \vec{u_y} \frac{\partial }{\partial y}+\vec{u_z} \frac{\partial }{\partial z}[/tex]

The del operator in matrix notation:

[tex]\nabla = \begin{pmatrix} \frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z} \end{pmatrix}[/tex] or [tex]\nabla = \begin{pmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} & \end{pmatrix}[/tex]

The divergence, here expressed in four different notations:

[tex]\nabla \cdot \vec{V}=\begin{pmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \end{pmatrix}\begin{pmatrix} V_x\\ V_y\\ V_z \end{pmatrix} =\sum_{a=\left \{ x,y,z \right \}}\frac{\partial V_a}{\partial a}=\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}[/tex]

The first expression, uses the del-dot operator, or a "nabla-dot" as LaTeX uses. The second expression is matrix multiplication. The third expression is a summation, as you sum over the terms as you let a=x, a=y, and a=z in turn. And the last expression is the fully expanded algebraic expression.

The third expression (summation notation) is the one that is closest to Einstein Notation, but you would replace x, y, z with x_1, x_2, x_3 or something like that, and somehow with the interplay of subscripts and superscripts, you imply summation, without actually bothering to put in the summation sign.

However, I'm not quite clear on the details, and I would benefit by seeing exactly what the Einstein Notation is for this case.

--------------------

The next operator is the "curl" or del-cross operator which is somewhat cumbersome in matrix form:

[tex]\nabla \times \vec {V} = \begin{pmatrix} \frac{\partial }{\partial y} & 0 & 0\\ 0 & \frac{\partial }{\partial z} & 0\\ 0 & 0 & \frac{\partial }{\partial x} \end{pmatrix} \begin{pmatrix} V_z\\ V_x\\ V_y \end{pmatrix}- \begin{pmatrix} \frac{\partial }{\partial z} & 0 & 0\\ 0 & \frac{\partial }{\partial x} & 0\\ 0 & 0 & \frac{\partial }{\partial y} \end{pmatrix} \begin{pmatrix} V_y\\ V_z\\ V_x \end{pmatrix}[/tex]

(I wonder whether there might be some simpler way to express it.)

This simplifies to

[tex]\nabla \times \vec {V} = \begin{pmatrix} \frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z}\\ \frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x}\\ \frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y} \end{pmatrix} =\left (\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z} \right )\vec {u_x}+ \left (\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x} \right )\vec {u_y}+ \left (\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y} \right )\vec {u_z}[/tex]

the last expression, is the expression given in the referenced e-book.

I have seen this expression given in two other ways:

[tex]\nabla \times \vec {V} = \det \begin{pmatrix} \vec u_x & \vec u_y & \vec u_z \\ \frac{\partial }{\partial x} &\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ V_x & V_y & V_z \end{pmatrix}[/tex]

which represents the cross-product of The vectors

[tex]\begin{pmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \end{pmatrix}[/tex]

and [tex]\begin{pmatrix} V_x & V_y & V_z \end{pmatrix}[/tex]

And in summation notation

[tex]\sum_{a=\left \{ x,y,z \right \}} \sum_{b=\left \{ x,y,z \right \}} \sum_{c=\left \{ x,y,z \right \}} \delta _{abc}\frac{\partial }{\partial a}V_b \vec u_c[/tex]

where

[tex]\delta_{abc}\overset{def}{=}\left\{\begin{matrix}

1,& abc=\left \{xyz,zyx,zxy \right \}\\

-1,& abc =\left \{zyx,yxz,xzy \right \}\\

0,& \mathrm a=b , b=c,c=a

\end{matrix}\right.[/tex]

Again, if I'm not mistaken, Einstein notation is most similar to the summation notation, but I'm not exactly sure what it would look like.

What I would like to do is give each of these formulas in three forms, and then ask a fairly simple question; What is the Einstein notation for each of these formulas?

The unit vectors, in matrix notation:

[tex]\vec{u_x}=\begin{pmatrix} 1\\ 0\\ 0 \end{pmatrix} or \begin{pmatrix} 1& 0& 0 \end{pmatrix}[/tex]

Similarly

[tex]\vec{u_y}=(0,1,0) , \vec{u_z}=(0,0,1)[/tex]

The "del" operator

[tex]\nabla = \vec{u_x} \frac{\partial }{\partial x} + \vec{u_y} \frac{\partial }{\partial y}+\vec{u_z} \frac{\partial }{\partial z}[/tex]

The del operator in matrix notation:

[tex]\nabla = \begin{pmatrix} \frac{\partial }{\partial x}\\ \frac{\partial }{\partial y}\\ \frac{\partial }{\partial z} \end{pmatrix}[/tex] or [tex]\nabla = \begin{pmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} & \end{pmatrix}[/tex]

The divergence, here expressed in four different notations:

[tex]\nabla \cdot \vec{V}=\begin{pmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \end{pmatrix}\begin{pmatrix} V_x\\ V_y\\ V_z \end{pmatrix} =\sum_{a=\left \{ x,y,z \right \}}\frac{\partial V_a}{\partial a}=\frac{\partial V_x}{\partial x}+\frac{\partial V_y}{\partial y}+\frac{\partial V_z}{\partial z}[/tex]

The first expression, uses the del-dot operator, or a "nabla-dot" as LaTeX uses. The second expression is matrix multiplication. The third expression is a summation, as you sum over the terms as you let a=x, a=y, and a=z in turn. And the last expression is the fully expanded algebraic expression.

The third expression (summation notation) is the one that is closest to Einstein Notation, but you would replace x, y, z with x_1, x_2, x_3 or something like that, and somehow with the interplay of subscripts and superscripts, you imply summation, without actually bothering to put in the summation sign.

However, I'm not quite clear on the details, and I would benefit by seeing exactly what the Einstein Notation is for this case.

--------------------

The next operator is the "curl" or del-cross operator which is somewhat cumbersome in matrix form:

[tex]\nabla \times \vec {V} = \begin{pmatrix} \frac{\partial }{\partial y} & 0 & 0\\ 0 & \frac{\partial }{\partial z} & 0\\ 0 & 0 & \frac{\partial }{\partial x} \end{pmatrix} \begin{pmatrix} V_z\\ V_x\\ V_y \end{pmatrix}- \begin{pmatrix} \frac{\partial }{\partial z} & 0 & 0\\ 0 & \frac{\partial }{\partial x} & 0\\ 0 & 0 & \frac{\partial }{\partial y} \end{pmatrix} \begin{pmatrix} V_y\\ V_z\\ V_x \end{pmatrix}[/tex]

(I wonder whether there might be some simpler way to express it.)

This simplifies to

[tex]\nabla \times \vec {V} = \begin{pmatrix} \frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z}\\ \frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x}\\ \frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y} \end{pmatrix} =\left (\frac{\partial V_z}{\partial y}-\frac{\partial V_y}{\partial z} \right )\vec {u_x}+ \left (\frac{\partial V_x}{\partial z}-\frac{\partial V_z}{\partial x} \right )\vec {u_y}+ \left (\frac{\partial V_y}{\partial x}-\frac{\partial V_x}{\partial y} \right )\vec {u_z}[/tex]

the last expression, is the expression given in the referenced e-book.

I have seen this expression given in two other ways:

[tex]\nabla \times \vec {V} = \det \begin{pmatrix} \vec u_x & \vec u_y & \vec u_z \\ \frac{\partial }{\partial x} &\frac{\partial }{\partial y} & \frac{\partial }{\partial z}\\ V_x & V_y & V_z \end{pmatrix}[/tex]

which represents the cross-product of The vectors

[tex]\begin{pmatrix} \frac{\partial }{\partial x} & \frac{\partial }{\partial y} & \frac{\partial }{\partial z} \end{pmatrix}[/tex]

and [tex]\begin{pmatrix} V_x & V_y & V_z \end{pmatrix}[/tex]

And in summation notation

[tex]\sum_{a=\left \{ x,y,z \right \}} \sum_{b=\left \{ x,y,z \right \}} \sum_{c=\left \{ x,y,z \right \}} \delta _{abc}\frac{\partial }{\partial a}V_b \vec u_c[/tex]

where

[tex]\delta_{abc}\overset{def}{=}\left\{\begin{matrix}

1,& abc=\left \{xyz,zyx,zxy \right \}\\

-1,& abc =\left \{zyx,yxz,xzy \right \}\\

0,& \mathrm a=b , b=c,c=a

\end{matrix}\right.[/tex]

Again, if I'm not mistaken, Einstein notation is most similar to the summation notation, but I'm not exactly sure what it would look like.

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