11. This extends the directional derivative of scalar functions to sections of vector bundles or principal bundles. 12. The precise definition of the directional derivative depends on what formalization one is using for vector fields. 13. This allows the abstraction of the notion of a directional derivative of a scalar function to general manifolds. 14. This operator applied to a function gives the directional derivative of at in the direction of increasing with fixed. 15. In differential geometry, the covariant derivative makes a choice for taking directional derivatives of vector fields along curves. 16. The Hopf lemma implies that the directional derivative of in the direction of the outward normal is strictly positive. 17. "' Hadamard derivative "'is a concept of directional derivative for maps between Banach spaces. 18. Multivariate functions can be handled with the same efficiency and mechanisms as univariate functions by adopting a directional derivative operator. 19. This means that the directional derivative depends only on the tangent vector of the curve at " p ". 20. The Fr�chet derivative should be contrasted to the more general G�teaux derivative which is a generalization of the classical directional derivative .
