Notation for Velocity Gradient

[swmmr1928]

This is not a problem statement this is not homework this is not a textbook exercise. This is my own question about a formula in a textbook.

Homework Statement

I am trying to understand the way that equation 4.2.2 is rewritten as equation 4.2.3

Homework Equations

Source: Stanley Middleman

The Attempt at a Solution

My confusion is with the 1/2 and the subscripts used.

 

[cepheid]

4.2.2. is NOT rewritten as 4.2.3. On the contrary, 4.2.3. is just an expression for the velocity gradient, a new thing that has been just introduced in the preceding description.

As for the factors of 1/2: just *expand* the stuff in parentheses on the right-hand side out, and you'll see that it is equal to the stuff on the left-hand side.

What, specifically, about the subscripts is confusing you? Basically both indices i and j can take on any of the values [1,2,3]. So you can think of 4.2.3 as a compact way of writing a whole SET of equations, one for each value of i and j.

 

[swmmr1928]

cepheid, yes of course-my mistake.

I would like to try to list these 9 equations to make sure I understand the notation.

SOrry, I don't have time to learn latex at the moment, but are the nine equations like this?:

equation 1: i = 1 j = 1
equation 2: i = 1 j = 2
equation 3: i = 1 j = 3
equation 4: i = 2 j = 1
equation 5: i = 2 j = 2
equation 6: i = 2 j = 3
equation 7: i = 3 j = 1
equation 8: i = 3 j = 2
equation 9: i = 3 j = 3

 

[cepheid]

cepheid, yes of course-my mistake.

I would like to try to list these 9 equations to make sure I understand the notation.

SOrry, I don't have time to learn latex at the moment, but are the nine equations like this?:

equation 1: i = 1 j = 1
equation 2: i = 1 j = 2
equation 3: i = 1 j = 3
equation 4: i = 2 j = 1
equation 5: i = 2 j = 2
equation 6: i = 2 j = 3
equation 7: i = 3 j = 1
equation 8: i = 3 j = 2
equation 9: i = 3 j = 3

Yeah, that looks about right to me.