The basis vectors are tangent the coordinate lines
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CHAPTER 2 |
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�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′ |
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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. 2-3 and Fig. 2-4) 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 ∂ | ∂ |
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| 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,
