covariant derivative

Remark 1: As we have seen in our articles Local Flatness or Local Inertial Frames and SpaceTime curvature and Local Inertial Frame (LIF), in a inertial frame of reference, the vanishing of the partial derivatives of the metric tensor at any point of M is equivalent to the vanishing of Christoffel symbols, and then we can write this fundamental equality in the context of any inertial or local inertial frame: Remark 2: the fact that the Christoffel symbol by itself does NOT transform as a tensor can be easily deduced from the fact that we can always find an (local) inertial frame in which its value equals zero, which should not be possible for a tensor.

, x��Z�n�F}߯�#0{�~�a �&�f��+(�c��юXW\?_]��Tqu[0G��E�$��~#X�i�EY�z��M%�,o�CXYw��l��_��S��ƊJ(ⴟ��K�{I�L}��n�5Ly�6ܴۛS��f*N��~�w;�u��7��v��7��I�O��v�o�|ߴ�c״��|||x��O��-�[� ��6BuY��MC��vWwMlĩ��C�>��}�wڻ;��g�&Z�ԝ�@#��vS1��/�a£��6��x(�Ċ�a[��!�y$a?�&�}��7c@B�C_GL� 7�W�ً%��ka��Fєo�#��L|�l� q� Roll this sheet of paper into a cylinder. ψ

D p X V The intesting property about the covariant derivative is that, as opposed to the usual directional derivative, this quantity transforms like a tensor, i.e. a 0

Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, Meinhard E. Mayer, "Principal Bundles versus Lie Groupoids in Gauge Theory", (1990) in, Review: David D. Bleecker, Gauge theory and variational principles, Geometrical aspects of local gauge symmetry, http://www.fuw.edu.pl/~dobaczew/maub-42w/node9.html, Gauge Principle For Ideal Fluids And Variational Principle, https://en.wikipedia.org/w/index.php?title=Gauge_covariant_derivative&oldid=936851540, Creative Commons Attribution-ShareAlike License, This page was last edited on 21 January 2020, at 11:47.

⟩ {\displaystyle \varphi }

To learn more, see our tips on writing great answers. In the general case, however, one must take into account the change of the coordinate system. (

a

p

x = $$\nabla_{j}A_{i}\equiv\frac{\partial A_{i}}{\partial x^{j}}{\color{Red} -}\Gamma_{{\color{Red} ij}}^{{\color{Red} k}}A_{k}$$ times the color symmetry Lie group SU(3). ϕ In 1950, Jean-Louis Koszul unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. As the contravector was arbitrary, this shows that U ∇ γ {\displaystyle \mathbf {v} } a

j

.

, ah yes! {\displaystyle \displaystyle \phi \,} {\displaystyle W^{j}} �\.�̉Q+�L��d1D��'��wD9�8S�ΕC��Dq�wnt��0��0��X�,�~P�D�ZV� gB�/rI�sA��"a��LT�s&Z3��h��s�Z��l2��! ;

∂ &= \partial_i (\mu_m u^m) - \mu_m(\partial_iu^m + \Gamma_{ij}^m u^j) \mathrm{\quad(by\ the\ formula\ given\ in\ your\ question)}\\

The covariant derivative on the other hand introduces its own change for vectors in a given direction, and it only depends on the vector direction at a single point, rather than a vector field in an open neighborhood of a point. This transformation law could serve as a starting point for defining the derivative in a covariant manner.

± �n^�]s ߠ^� ��]�)��p`�8��Pqp��3l�N`�g0��̞ (��{��=#>�Ej� ]nI�آWs��z"�=��M�uF`��9 'R��mfj�^?�e�ܫ�l* Q��#mD�sE��1�1By $$\mathbf{A} = A^{1}\mathbf{e_{1}}+A^{2}\mathbf{e_{2}}$$ t Consider a vector V = Vαeα (ie the tensor has contravariant components Vα and coordinate basis vectors eα). [1][2][3] Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection. S

The infinitesimal change of the vector is a measure of the curvature.

and

The intesting property about the covariant derivative is that, as opposed to the usual directional derivative, this quantity transforms like a tensor, i.e. This article attempts to hew most closely to the notation and language commonly employed in physics curriculum, touching only briefly on the more abstract connections. b

Now consider how all of this plays out in the context of general relativity.

we get, The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field u.
. ψ {\displaystyle (\mathbf {e} _{r},\mathbf {e} _{\theta })} a k Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.)

$$A= \sum_{i=1}^{n}A^{i}\mathbf{e_{i}}$$, Taking the derivative wrt the tangent basis vector and dropping the summation by Einstein convention: ,

is the function that associates with each point p in the common domain of f and v the scalar t } ; Let $u^m$ be a contravector and $\mu_m$ a covector. It can be expressed in the following form:[12]. [6] Using ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones.

, acting on a field