What does the j means in this notation?

[sams]

This section of Introduction to Electrodynamics by Griffiths, section 8.2.2 (page 363), is talking about the Maxwell's Stress Tensor. I do not quite understand what the j means on the left = sign (for either of the two representations) in the attached figure highlighted in yellow color. I have read about tensors and their Index Gymnastics but I still do not understand this notation. Any help is much appreciated.

Many thanks!

 

[Orodruin]

The LHS is the jth component of the vector $\vec a \cdot \overleftrightarrow T$.

 

[sams]

How can j affect the summation on the right side of the equation? Does it decrease the tensor by one rank?

 

[Orodruin]

Each of the equations is really three different equations, one for every possible value of j.

 

[sams]

How come they are different equations? Isn't it one equation, since we have the summation operator leading to different terms but added together to form one equation?

 

[Orodruin]

You do not have a summation over j. The j represents different components of a vector. If you take two vectors $\vec v$ and $\vec w$ that are equal, this is the same as saying that their components are equal. This would be written $v_j = w_j$, which represents one equation for every possible value of $j$. The equations you posted are really no different from this. They are telling you how the components of $\vec a \cdot \overleftrightarrow T$ and $\overleftrightarrow T \cdot \vec a$ relate to the components of $\vec a$ and $\overleftrightarrow T$.

 

[Orodruin]

To be more specific. Writing out the sum in the first equation would give you
$$
\sum_{i = x,y,z} a_i T_{ij} = a_x T_{xj} + a_y T_{yj} + a_z T_{zj}.
$$
Letting $j = x$ you would now get
$$
\left(\vec a \cdot \overleftrightarrow T\right)_x = a_x T_{xx} + a_y T_{yx} + a_z T_{zx}
$$
whereas if you let $j = y$ you get the different relation
$$
\left(\vec a \cdot \overleftrightarrow T\right)_y = a_x T_{xy} + a_y T_{yy} + a_z T_{zy}
$$
and so on.

 

[sams]

Thank you so much Orodruin for the valuable information...