directional derivative in a sentence

Examples

1. The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor.
2. where \ mathbf { B } \ cdot \ nabla is the directional derivative in the direction of \ mathbf { B } multiplied by its magnitude.
3. That is, is the unique 1-form such that for every smooth vector field,, where is the directional derivative of in the direction of.
4. Often we mean the directional derivative in the direction of a vector field; this specializes to the standard case of the derivative in terms of one coordinate.
5. Further, am I correct in asserting that taking that " inner product " of the Jacobian with a vector produces a directional derivative of the vector field?
6. Note that existence of the partial derivatives ( or even all of the directional derivatives ) does not in general guarantee that a function is differentiable at a point.
7. In mathematics, the "'directional derivative "'of a multivariate differentiable function along a given curvilinear coordinate curves, all other coordinates being constant.
8. This is analogous to the fact that the existence of all directional derivatives at a point does not guarantee total differentiability ( or even continuity ) at that point.
9. The tangent space of a point consists of the possible directional derivatives at that point, and has the same dimension " n " as does the manifold.
10. For manifolds that are subsets of "'R " "'n ", this tangent vector will agree with the directional derivative defined above.