The basis vectors are tangent the coordinate lines

CHAPTER 2 

�Ai Bi→ Ai Bi= A1B1+ A2B2+ A3B3
Another example is that SaTab is shorthand for the expression
Aa′= �a′bAb
In this expression, b is once again a dummy index. The sum implied here means that
Aa′= �a′  bAb = �a′  0A0 + �a′  1A1 + �a′  2A2 + �a′ 

would be valid to make the change a′→ b′, provided that we make this change on both sides; i.e.,
Ab′= �b′bAb
In a given coordinate system, the basis vectors ea are tangent to the coordinate lines. (See Fig. 23 and Fig. 24) This is the reason why we can write basis vectors as partial derivatives in a particular coordinate direction (for an explanation, see Carroll, 2004). In other words, we take
e ∂  ∂ 

a =a =  ∂xa 
This type of basis is called a coordinate basis. This allows us to think of a vector as an operator, one that maps a function into a new function that is related to its derivative. In particular,