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The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative, the covariant derivative is a rule, , which takes as its inputs: (1) a vector, u, defined at a point P, and (2) a vector field v defined in a neighborhood of P. [7]
The covariant derivative of a function ... The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero:
The symmetry of the Christoffel symbol now implies = for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (see curvature tensor). The covariant derivative of a type (2, 0) tensor field A ik is = + +, that is, ; =, + +.
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics.These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.
First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin. Then f and its partial derivatives with respect to x and y vanish at (0,0). Therefore, the Taylor expansion of f at (0,0) starts with quadratic terms:
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form.
When the torsion tensor is zero, so that [,] =, we may use this fact to write Riemann curvature tensor as [2] (,) =,,. Similarly, one may also obtain the second covariant derivative of a function f as
The exterior derivative of a totally antisymmetric type (0, s) tensor field with components A α 1 ⋅⋅⋅α s (also called a differential form) is a derivative that is covariant under basis transformations. It does not depend on either a metric tensor or a connection: it requires only the structure of a differentiable manifold.