Euclidean Vector - Vectors As Directional Derivatives

Vectors As Directional Derivatives

A vector may also be defined as a directional derivative: consider a function and a curve . Then the directional derivative of is a scalar defined as

where the index is summed over the appropriate number of dimensions (for example, from 1 to 3 in 3-dimensional Euclidean space, from 0 to 3 in 4-dimensional spacetime, etc.). Then consider a vector tangent to :

The directional derivative can be rewritten in differential form (without a given function ) as

Therefore any directional derivative can be identified with a corresponding vector, and any vector can be identified with a corresponding directional derivative. A vector can therefore be defined precisely as

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