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.
32.
Where \ mathbf { B } \ cdot \ nabla is the directional derivative in the direction of \ mathbf { B } multiplied by its magnitude.
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That is, is the unique 1-form such that for every smooth vector field,, where is the directional derivative of in the direction of.
34.
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.
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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?
36.
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.
37.
In mathematics, the "'directional derivative "'of a multivariate differentiable function along a given curvilinear coordinate curves, all other coordinates being constant.
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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.
39.
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.
40.
For manifolds that are subsets of "'R " "'n ", this tangent vector will agree with the directional derivative defined above.
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