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Covariant derivative
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==Formal definition== A covariant derivative is a [[connection (vector bundle)|(Koszul) connection]] on the [[tangent bundle]] and other [[tensor bundle]]s: it differentiates vector fields in a way analogous to the usual differential on functions. The definition extends to a differentiation on the dual of vector fields (i.e. [[cotangent space|covector]] fields) and to arbitrary [[tensor field]]s, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction). ===Functions=== Given a point <math>p \in M</math> of the manifold {{mvar|M}}, a real function <math>f : M \to \R</math> on the manifold and a tangent vector <math>\mathbf{v} \in T_pM</math>, the covariant derivative of {{mvar|f}} at {{mvar|p}} along {{math|'''v'''}} is the scalar at {{mvar|p}}, denoted <math>\left(\nabla_\mathbf{v} f\right)_p</math>, that represents the [[Principal part#Calculus|principal part]] of the change in the value of {{mvar|f}} when the argument of {{mvar|f}} is changed by the infinitesimal displacement vector {{math|'''v'''}}. (This is the [[differential of a function|differential]] of {{mvar|f}} evaluated against the vector {{math|'''v'''}}.) Formally, there is a differentiable curve <math>\phi:[-1, 1]\to M</math> such that <math>\phi(0) = p</math> and <math>\phi'(0) = \mathbf{v}</math>, and the covariant derivative of {{mvar|f}} at {{mvar|p}} is defined by <math display="block">\left(\nabla_\mathbf{v} f\right)_p = \left(f \circ \phi\right)^\prime \left(0\right) = \lim_{t \to 0} \frac{ f(\phi\left(t\right)) - f(p) }{t}.</math> When <math>\mathbf{v} : M \to T_pM</math> is a vector field on {{mvar|M}}, the covariant derivative <math>\nabla_\mathbf{v}f : M \to \R </math> is the function that associates with each point {{mvar|p}} in the common domain of {{mvar|f}} and {{math|'''v'''}} the scalar <math>\left(\nabla_\mathbf{v}f\right)_p</math>. For a scalar function {{mvar|f}} and vector field {{math|'''v'''}}, the covariant derivative <math>\nabla_\mathbf{v} f</math> coincides with the [[Lie derivative]] <math>L_v(f)</math>, and with the [[exterior derivative]] <math>df(v)</math>. ===Vector fields=== Given a point {{mvar|p}} of the manifold {{mvar|M}}, a vector field <math>\mathbf{u} : M \to T_p M</math> defined in a neighborhood of {{mvar|p}} and a tangent vector <math>\mathbf{v} \in T_pM</math>, the covariant derivative of {{math|'''u'''}} at {{mvar|p}} along {{math|'''v'''}} is the tangent vector at {{mvar|p}}, denoted <math>(\nabla_\mathbf{v} \mathbf{u})_p</math>, such that the following properties hold (for any tangent vectors {{math|'''v'''}}, {{math|'''x'''}} and {{math|'''y'''}} at {{mvar|p}}, vector fields {{math|'''u'''}} and {{math|'''w'''}} defined in a neighborhood of {{mvar|p}}, scalar values {{mvar|g}} and {{mvar|h}} at {{mvar|p}}, and scalar function {{mvar|f}} defined in a neighborhood of {{mvar|p}}): # <math>\left(\nabla_\mathbf{v} \mathbf{u}\right)_p</math> is linear in <math>\mathbf{v}</math> so <math display="block">\left(\nabla_{g\mathbf{x} + h\mathbf{y}} \mathbf{u}\right)_p = g(p) \left(\nabla_\mathbf{x} \mathbf{u}\right)_p + h(p) \left(\nabla_\mathbf{y} \mathbf{u}\right)_p</math> # <math>\left(\nabla_\mathbf{v} \mathbf{u}\right)_p</math> is additive in <math>\mathbf{u}</math> so: <math display="block">\left(\nabla_\mathbf{v}\left[\mathbf{u} + \mathbf{w}\right]\right)_p = \left(\nabla_\mathbf{v} \mathbf{u}\right)_p + \left(\nabla_\mathbf{v} \mathbf{w}\right)_p</math> # <math>(\nabla_\mathbf{v} \mathbf{u})_p</math> obeys the [[product rule]]; i.e., where <math>\nabla_\mathbf{v}f</math> is defined above, <math display="block">\left(\nabla_\mathbf{v} \left[f\mathbf{u}\right]\right)_p = f(p)\left(\nabla_\mathbf{v} \mathbf{u})_p + (\nabla_\mathbf{v}f\right)_p\mathbf{u}_p.</math> Note that <math>\left(\nabla_\mathbf{v} \mathbf{u}\right)_p</math> depends not only on the value of {{math|'''u'''}} at {{mvar|p}} but also on values of {{math|'''u'''}} in a neighborhood of {{mvar|p}}, because the last property, the product rule, involves the directional derivative of {{mvar|f}} (by the vector {{math|'''v'''}}). If {{math|'''u'''}} and {{math|'''v'''}} are both vector fields defined over a common domain, then <math>\nabla_\mathbf{v}\mathbf u</math> denotes the vector field whose value at each point {{mvar|p}} of the domain is the tangent vector <math>\left(\nabla_\mathbf{v}\mathbf u\right)_p</math>. ===Covector fields=== Given a field of [[Cotangent space|covectors]] (or [[one-form]]) <math>\alpha</math> defined in a neighborhood of {{mvar|p}}, its covariant derivative <math>(\nabla_\mathbf{v}\alpha)_p</math> is defined in a way to make the resulting operation compatible with tensor contraction and the product rule. That is, <math>(\nabla_\mathbf{v}\alpha)_p</math> is defined as the unique one-form at {{mvar|p}} such that the following identity is satisfied for all vector fields {{math|'''u'''}} in a neighborhood of {{mvar|p}} <math display="block">\left(\nabla_\mathbf{v}\alpha\right)_p \left(\mathbf{u}_p\right) = \nabla_\mathbf{v}\left[\alpha\left(\mathbf{u}\right)\right]_p - \alpha_p\left[\left(\nabla_\mathbf{v}\mathbf{u}\right)_p\right].</math> The covariant derivative of a covector field along a vector field {{math|'''v'''}} is again a covector field. ===Tensor fields=== Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary [[Tensor (intrinsic definition)|tensor]] fields by imposing the following identities for every pair of tensor fields <math> \varphi</math> and <math>\psi </math> in a neighborhood of the point {{mvar|p}}: <math display="block">\nabla_\mathbf{v}\left(\varphi \otimes \psi\right)_p = \left(\nabla_\mathbf{v}\varphi\right)_p \otimes \psi(p) + \varphi(p) \otimes \left(\nabla_\mathbf{v}\psi\right)_p,</math> and for <math>\varphi</math> and <math>\psi</math> of the same valence <math display="block">\nabla_\mathbf{v}(\varphi + \psi)_p = (\nabla_\mathbf{v}\varphi)_p + (\nabla_\mathbf{v}\psi)_p.</math> The covariant derivative of a tensor field along a vector field {{math|'''v'''}} is again a tensor field of the same type. Explicitly, let {{mvar|T}} be a tensor field of type {{math|(''p'', ''q'')}}. Consider {{mvar|T}} to be a differentiable [[multilinear map]] of [[smooth function|smooth]] [[section (fiber bundle)|sections]] {{math|''Ξ±''{{isup|1}}, ''Ξ±''{{isup|2}}, ..., ''Ξ±''<sup>''q''</sup>}} of the cotangent bundle {{math|''T''{{isup|β}}''M''}} and of sections {{math|''X''{{sub|1}}, ''X''{{sub|2}}, ..., ''X''<sub>''p''</sub>}} of the [[tangent bundle]] {{math|''TM''}}, written {{math|''T''(''Ξ±''{{isup|1}}, ''Ξ±''{{isup|2}}, ..., ''X''{{sub|1}}, ''X''{{sub|2}}, ...)}} into {{math|'''R'''}}. The covariant derivative of {{mvar|T}} along {{mvar|Y}} is given by the formula <math display="block">\begin{align} (\nabla_Y T)\left(\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots\right) = &{} \nabla_Y\left(T\left(\alpha_1,\alpha_2, \ldots, X_1, X_2, \ldots\right)\right) \\ &{}- T\left(\nabla_Y\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots\right) - T\left(\alpha_1, \nabla_Y\alpha_2, \ldots, X_1, X_2, \ldots\right) - \cdots \\ &{}- T\left(\alpha_1, \alpha_2, \ldots, \nabla_YX_1, X_2, \ldots\right) - T\left(\alpha_1, \alpha_2, \ldots, X_1, \nabla_Y X_2, \ldots\right) - \cdots \end{align}</math>
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