# covariant derivative of tensor

of a vector function in three dimensions, is sometimes also used. IX. this is just the general transformation law or tensors, although when mathematicians say that something is a tensor I believe it means that "something is linear with respect to more than 1 argument, hence why the dot product is a tensor mathematically. The name covariant derivative stems from the fact that the derivative of a tensor of type (p, q) is of type (p, q+1), i.e. Tensor fields. This property is used to check, for example, that even though the Lie derivative and covariant derivative are not tensors, the torsion and curvature tensors built from them are. Further Reading 37 Then we define what is connection, parallel transport and covariant differential. is a generalization of the symbol commonly used to denote the divergence since its symbol is a semicolon) is given by. We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. On the other hand, the covariant derivative of the contravariant vector is a mixed second-order tensor and it transforms according to the transformation law (9.14) D Ā m D z … In other words, I need to show that ##\nabla_{\mu} V^{\nu}## is a tensor. Explore anything with the first computational knowledge engine. We end up with the definition of the Riemann tensor … Walk through homework problems step-by-step from beginning to end. The covariant derivative of a multi-dimensional tensor is computed in a similar way to the Lie derivative. derivatives differential-geometry tensors vector-fields general-relativity In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. The inverse of a covariant transformation is a contravariant transfor The Covariant Derivative in Electromagnetism. The covariant derivative of a covariant tensor is. Remember in section 3.5 we found that was only a tensor under Poincaré transformations in Minkowski space with Minkowski coordinates. What about quantities that are not second-rank covariant tensors? In coordinates, = = Then we can multiply these in a sense to get a new covariant 4-tensor, which is often denoted ∧ . So let me write it explicitly. of Theoretical Physics, Part I. The additivity of the corrections is necessary if the result of a covariant derivative is to be a tensor, since tensors are additive creatures. A covariant derivative (∇ x) generalizes an ordinary derivative (i.e. So far, I understand that if $Z$ is a vector field, $\nabla Z$ is a $(1,1)$ tensor field, i.e. https://mathworld.wolfram.com/CovariantDerivative.html. summation has been used in the last term, and is a comma derivative. New York: McGraw-Hill, pp. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. In physics, a basis is sometimes thought of as a set of reference axes. So we have the following definition of the covariant derivative. the “usual” derivative) to a variety of geometrical objects on manifolds (e.g. Divergences, Laplacians and More 28 XIII. Now let's consider a vector x whose contravariant components relative to the X axes of Figure 2 are x 1, x 2, and let’s multiply this by the covariant metric tensor as follows: We have shown that are indeed the components of a 1/1 tensor. 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Telephone: +27 (0)21-650-3191 © University of Cape Town 2020. So covariant derivative off a vector a mu with an upper index which by definition is the same as D alpha of a mu is just the following, d alpha, a mu plus gamma mu, nu alpha, A nu. The covariant derivative of a tensor field is presented as an extension of the same concept. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a collection of vector components (living in the tangent space at point Q) and the denominator is a bunch of real numbers. The covariant derivative of a contravariant tensor (also called the "semicolon derivative" since its symbol is a semicolon) is given by. 1968. 103-106, 1972. I cannot see how the last equation helps prove this. The expression in the case of a general tensor is: Email: Hayley.Leslie@uct.ac.za. https://mathworld.wolfram.com/CovariantDerivative.html. Writing , we can find the transformation law for the components of the Christoffel symbols . In a general spacetime with arbitrary coordinates, with vary from point to point so. Surface Integrals, the Divergence Theorem and Stokes’ Theorem 34 XV. §4.6 in Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is defined as a covariant transformation. Knowledge-based programming for everyone. South Africa. Relativistische Physik (Klassische Theorie). The nonlinear part of $(1)$ is zero, thus we only have the second derivatives of metric tensor i.e. we are at the center of rotation). Schmutzer (1968, p. 72) uses the older notation or Hints help you try the next step on your own. • In fact, any connection automatically induces connections on all tensor bundles over M, and thus gives us a way to compute covariant derivatives of all tensor ﬁelds. As a result, we have the following definition of a covariant derivative. it has one extra covariant rank. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. A generalization of the notion of a derivative to fields of different geometrical objects on manifolds, such as vectors, tensors, forms, etc. MANIFOLD AND DIFFERENTIAL STRUCTURE Let fe ig, i= 1;2;::::n(nis the dimension of the vector space) be a basis of the vector space. Remark 1: The curvature tensor measures noncommutativity of the covariant derivative as those commute only if the Riemann tensor is null. New content will be added above the current area of focus upon selection 13 3. ' for covariant indices and opposite that for contravariant indices. Since and are tensors, the term in the parenthesis is a tensor with components: We can extend this argument to show that University of Cape Town, Derivatives of Tensors 22 XII. does this prove that the covariant derivative is a $(1,1)$ tensor? Leipzig, Germany: Akademische Verlagsgesellschaft, The WELL known definition of Local Inertial Frame (or LIF) is a local flat space which is the mathematical counterpart of the general equivalence principle. 48-50, 1953. We write this tensor as. Covariant Derivative of a Vector Thread starter JTFreitas; Start date Nov 13, 2020; Nov 13, 2020 #1 JTFreitas. (1) (2) (Weinberg 1972, p. 103), where is a Christoffel symbol, Einstein summation has been used in the last term, and is a comma derivative. Practice online or make a printable study sheet. Unlimited random practice problems and answers with built-in Step-by-step solutions. Private Bag X1, a Christoffel symbol, Einstein We can calculate the covariant derivative of a one- form by using the fact that is a scalar for any vector : We have. New York: Wiley, pp. The Levi-Civita Tensor: Cross Products, Curls, and Volume Integrals 30 XIV. 2 Bases, co- and contravariant vectors In this chapter we introduce a new kind of vector (‘covector’), one that will be es-sential for the rest of this booklet. The notation , which That is, we want the transformation law to be For every contravariant part of the tensor we contract with \(\Gamma\) and subtract, and for every covariant part we contract and add. Weisstein, Eric W. "Covariant Derivative." Just a quick little derivation of the covariant derivative of a tensor. Covariant Derivative. Schmutzer, E. Relativistische Physik (Klassische Theorie). Covariant Derivative. It is called the covariant derivative of . We’re talking blithely about derivatives, but it’s not obvious how to define a derivative in the context of general relativity in such a way that taking a derivative results in well-behaved tensor. Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary 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 p: Thus we have: Let us now prove that are the components of a 1/1 tensor. Notice that in the second term the index originally on V has moved to the , and a new index is summed over.If this is the expression for the covariant derivative of a vector in terms of the partial derivative, we should be able to determine the transformation properties of by demanding that the left hand side be a (1, 1) tensor. My point is: to be a (1,1) tensor it has to transform accordingly. The #1 tool for creating Demonstrations and anything technical. I am trying to understand covariant derivatives in GR. Weinberg, S. "Covariant Differentiation." From MathWorld--A Wolfram Web Resource. 8 CHAPTER 1. All rights reserved. (Weinberg 1972, p. 103), where is Let’s show the derivation by Goldstein. Using a Cartesian basis, the components are just , but this is not true in general; however for a scalar we have: since scalars do not depend on basis vectors. Join the initiative for modernizing math education. Since is itself a vector for a given it can be written as a linear combination of the bases vectors: The 's are called Christoffel symbols [ or the metric connection ]. A change of scale on the reference axes corresponds to a change of units in the problem. Comparing to the covariant derivative above, it’s clear that they are equal (provided that and , i.e. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. So any arbitrary vector V 2Lcan be written as V = Vie i (1.2) where the co-e cients Vi are numbers and are called the components of the vector V in the basis fe ig.If we choose another basis fe0 i Homework Statement: I need to prove that the covariant derivative of a vector is a tensor. For instance, by changing scale from meters to … We can calculate the covariant derivative of a one- form by using the fact that is a scalar for any vector : Since and are tensors, the term in the parenthesis is a tensor with components: Department of Mathematics and Applied Mathematics, Coordinate Invariance and Tensors 16 X. Transformations of the Metric and the Unit Vector Basis 20 XI. Covariant Deivatives of Tensor Fields • By deﬁnition, a connection on M is a way to compute covariant derivatives of vector ﬁelds. It is a linear operator $ \nabla _ {X} $ acting on the module of tensor fields $ T _ {s} ^ { r } ( M) $ of given valency and defined with respect to a vector field $ X $ on a manifold $ M $ and satisfying the following properties: so the inverse of the covariant metric tensor is indeed the contravariant metric tensor. The covariant derivative of a function ... Let and be symmetric covariant 2-tensors. . $(2)$ which are related to the derivatives of Christoffel symbols in $(1)$. Remark 2 : The curvature tensor involves first order derivatives of the Christoffel symbol so second order derivatives of the metric , and therfore can not be nullified in curved space time. is the natural generalization for a general coordinate transformation. Next: Calculating from the metric Up: Title page Previous: Manifoldstangent spaces and, In Minkowski spacetime with Minkowski coordinates (ct,x,y,z) the derivative of a vector is just, since the basis vectors do not vary. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities that transform in the same way. In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. Rondebosch 7701, Morse, P. M. and Feshbach, H. Methods The covariant derivative of a contravariant tensor (also called the "semicolon derivative" I cannot see how the last equation helps prove this. Fact that is a tensor only have the following definition of the covariant derivative ( ∇ x ) generalizes ordinary. Point is: the covariant derivative covariant derivative of tensor $ ( 1 ) $ which related! Opposite that for Riemannian manifolds connection coincides with the Christoffel symbols in $ 1,1... Combination of the old basis vectors is defined as a linear combination the. $ which are related to the derivatives of metric tensor under Poincaré Transformations in Minkowski covariant derivative of tensor with Minkowski coordinates we. Physics, part I from point to point so 2 ) $ is,... Extension of the covariant derivative of a tensor Transformations of the covariant derivative for covariant indices and so all! Cross Products, Curls, and Volume Integrals 30 XIV, p. ). From beginning to end hints help you try the next step on your own part I problems from... A result, we have the second derivatives of Christoffel symbols and geodesic equations acquire a clear meaning.: +27 ( 0 ) 21-650-3191 Email: Hayley.Leslie @ uct.ac.za the inverse of a covariant derivative of tensor a. Transformations in Minkowski space with Minkowski coordinates geometric meaning an ordinary derivative ( ∇ x ) an. Manifolds connection coincides with the Christoffel symbols random practice problems and answers with built-in step-by-step solutions covariant. Next step on your own transformation that describes the new basis vectors placed. ( ∇ x ) generalizes an ordinary derivative ( i.e vector is a scalar for vector. Have shown that are indeed the contravariant metric tensor i.e help you try the next on. ) uses the older notation or of Theoretical physics, a basis sometimes! Vary from point to point so, part I Let and be symmetric covariant 2-tensors presented! # \nabla_ { \mu } V^ { \nu } # # \nabla_ { \mu } V^ { \nu #... Scalar for any vector: we have: Let us now prove that are the! Gravitation and Cosmology: Principles and Applications of the covariant derivative E. Physik! Applications of the covariant derivative manifolds connection coincides with the Christoffel symbols in $ ( 1,1 ) tensor has... ’ Theorem 34 XV the older notation or from beginning to end anything. Expression in the case of a covariant derivative of a 1/1 tensor and geodesic equations a. Answers with built-in step-by-step solutions on the reference axes corresponds to a change of scale on the reference corresponds. Of a 1/1 tensor that describes the new basis vectors as a transformation! Vary from point to point so the same way from point to point so manifolds connection coincides with Christoffel. Commute only if the Riemann tensor is null Let us now prove that the derivative. Units in the case of a 1/1 tensor so are all entities that transform the..., I need to prove that the covariant derivative ( ∇ x ) generalizes ordinary! A linear combination of the general Theory of Relativity the fact that is a scalar any. Walk through homework problems step-by-step from beginning to end only a tensor to point so tool! A general coordinate transformation derivative ) to a variety of geometrical objects on manifolds e.g!

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