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**Difficulty with notation**

I've always had trouble with notation, since there are so many ways to write the same thing.

This one is for Vector Analysis

First off, when speaking of unit vectos, such as

**i, j, k**is it ok to use the notation;

[tex]

{\hat i},{\hat j},{\hat k}

[/tex]?

Such as in

[tex]

f(x,y,z) = (z^3+2y){\hat i}+(3x^2+5y){\hat j}...

[/tex]

Or is this ment to notate something else?

Secondly, in the folowing, does the [itex]{\varepsilon _{{\bf i,j,k}} }[/itex] (I'm noting now that this is known as the permutation symbol or Levi-Civita density) represent the same thing as [itex]{{\bf e} _{{\bf i,j,k}} }[/itex] or [itex]{{\bf e} _{1,2,3} }[/itex] (unit vectors)?

[tex]

\sum\limits_{{\bf i,j,k}} {\varepsilon _{{\bf i,j,k}} } A_{\bf i} B_{\bf j} C_{\bf k}

[/tex]

Also, I am not quite sure how to expand this. I am trying to prove that

[tex]

\vec A\vec B\vec C= \sum\limits_{{\bf i,j,k}} {\varepsilon _{{\bf i,j,k}} } A_{\bf i} B_{\bf j} C_{\bf k}

[/tex]

The left side is easy to expand through a matrix. Maybe once I understand the notation the rest will make sense.

Is this the complete expansion:

[tex]

\varepsilon _{{\bf i,j,k}} A_{\bf i} B_{\bf j} C_{\bf k} + \varepsilon _{{\bf k,i,j}} A_{\bf k} B_{\bf i} C_{\bf j} + \varepsilon _{{\bf j,k,i}} A_{\bf j} B_{\bf k} C_{\bf i} + \varepsilon _{{\bf i,k,j}} A_{\bf i} B_{\bf k} C_{\bf j} + \varepsilon _{{\bf j,i,k}} A_{\bf j} B_{\bf i} C_{\bf k} + \varepsilon _{{\bf k,j,i}} A_{\bf j} B_{\bf i} C_{\bf k}

[/tex]

Or is there really also all the zero tems?

[tex]

\varepsilon _{{\bf i,i,k}} A_{\bf i} B_{\bf i} C_{\bf k}...

[/tex]

I think there is 12 total.

Then the even permutation is equal to 1 and the odd is equal to -1 so:

[tex]

A_{\bf i} B_{\bf j} C_{\bf k} + A_{\bf k} B_{\bf i} C_{\bf j} + A_{\bf j} B_{\bf k} C_{\bf i} - A_{\bf i} B_{\bf k} C_{\bf j} - A_{\bf j} B_{\bf i} C_{\bf k} - A_{\bf j} B_{\bf i} C_{\bf k}

[/tex]

Which is exactly the triple product expansion

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