# covariant derivative of a vector

$$Pay attention because you'll see that this pushforward \Sigma_\ast Z is what will make the additional t^\nu_B appear in your middle term. A' A A'q A'r dq Q: Which of A a!$$, $$The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination $\Gamma^k \mathbf{e}_k\,$. I was bitten by a kitten not even a month old, what should I do? Now you want to understand differentiation of t^\mu_A. This will be useful for defining the accelerationof a curve, which is the covariant derivative of the velocity vector with respect to itself, and for defining geodesics, which are curves with zero acceleration. In the scalar case ∇φ is simply the gradient of a scalar, while ∇A is the covariant derivative of the macroscopic vector (which can also be thought of as the Jacobian matrix of A as a function of x). An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction. where \pi is the bundle projection. Is a password-protected stolen laptop safe? is a scalar density of weight 1, and is a scalar density of weight w. (Note that is a density of weight 1, where is the determinant of the metric. A vector field $${w}$$ on $${M}$$ can be viewed as a vector-valued 0-form. We can form their tensor product \Sigma^\ast(TM)\otimes T^\ast W and endow it with a connection (\Sigma^\ast \nabla)\otimes D defined to act on tensor products of sections:$$(\Sigma^\ast \nabla\otimes D)_Z(f\otimes g)=(\Sigma^\ast \nabla_Z f)\otimes g+ f\otimes (D_Z g).$$. Therefore consider$${\frak t}=t^\mu_A (\Sigma^\ast \partial_\mu)\otimes d\xi^A.$$. is the metric, and are the Christoffel symbols.. is the covariant derivative, and is the partial derivative with respect to .. is a scalar, is a contravariant vector, and is a covariant vector. Easily Produced Fluids Made Before The Industrial Revolution - Which Ones? is it possible to read and play a piece that's written in Gflat (6 flats) by substituting those for one sharp, thus in key G? Often, vectors i.e., elements of the vector space Lare called contravariant vectors and elements of dual space L, i.e., the covectors are called covariant vectors. What to do? The connection must have either spacetime indices or world sheet indices. As noted previously, the covariant derivative $${\nabla_{v}w}$$ is linear in $${v}$$ and depends only on its local value, and so can be viewed as a vector … Focusing in your case, it is defined to be$$\Sigma^\ast (TM)=\{(\xi,v)\in W\times TM : v\in T_{\Sigma(\xi)}M\},\quad \pi(\xi,v)=\xi,$$. Each of the D fields (one for each value of \mu) will transform as a diffeomorphism scalar and its index \mu plays no role on the transformation. To see what it must be, consider a basis B = {e α} defined at each point on the manifold and a vector field v α which has constant components in basis B. A covariant derivative is a tensor which reduces to a partial derivative of a vector field in Cartesian coordinates. But how to imagine visually the covariant derivative of tangent vectors. MathJax reference. Now if E_a is a local frame in M in a neighborhood of \Sigma(W), since {\cal S}(\xi)\in T_{\Sigma(\xi)}M we can always expand$${\cal S}(\xi)={\cal S}^a(\xi)E_a(\Sigma(\xi))$$, and therefore a section S : W\to \Sigma^\ast(TM) is always expanded in a basis of pullback sections$$S(\xi)={\cal S}^a(\xi) \Sigma^\ast E_a(\xi)$$, All this construction allows that a connection \nabla on TM naturally induce a connection \Sigma^\ast \nabla on \Sigma^\ast(TM). The covariant derivative is a generalization of the directional derivative from vector calculus. The second term, however, demands us to evaluate \Sigma_\ast Z. Expert Answer . \gamma^{C}_{AB}=\frac{1}{2}\gamma^{CD}(\gamma_{DA,B}+\gamma_{DB,A}-\gamma_{AB,D}) \Gamma^{\mu}_{\kappa\lambda}=\frac{1}{2}g^{\mu\nu}(g_{\nu\kappa,\lambda}+g_{\nu\lambda,\kappa}-g_{\kappa\lambda,\nu}) How to write complex time signature that would be confused for compound (triplet) time? Lemma 8.1 (Projection onto the Tangent Space) Now you have a metric g on M. You can see a vector field. Now let's consider a vector x whose contravariant components relative to the X axes of Figure 2 are x 1, x 2, and let’s multiply this by the covariant metric tensor as follows: Remember that summation is implied over the repeated index u, whereas the index v appears only once (in any given product) so this expression applies for any value of v. That is, we want the transformation law to be For a scalar, the covariant derivative is the same as the partial derivative, and is … The covariant derivative of the r component in the q direction is the regular derivative plus another term. The covariant derivative of the r component in the q direction is the regular derivative plus another term. It only takes a minute to sign up.$$ First we cover formal definitions of tangent vectors and then proceed to define a means to “covariantly differentiate”. Exterior covariant derivative for vector bundles When ρ : G → GL(V) is a representation, one can form the associated bundle E = P × ρ V.Then the exterior covariant derivative D given by a connection on P induces an exterior covariant derivative (sometimes called the exterior connection) on the associated bundle, this time using the nabla symbol: This is following Lee’s Riemannian Manifolds, … I'm going to propose an approach to justify the formula in the OP employing the idea of pullback bundles and pullback connections. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. If a vector field is constant, then Ar ;r=0. This question hasn't been answered yet Ask an expert. D_{B} t^{\mu}_A=t^{\mu}_{A},_B+ \Gamma^{\mu}_{\kappa\lambda}t^{\kappa}_{A}t^{\lambda}_B-\Gamma^C_{AB}t^{\mu}_C First you should ask what this is as an intrinsic object. Should we leave technical astronomy questions to Astronomy SE? 1 < i,j,k < n, then defining the covariant derivative of a vector field by the above formula, we obtain an affine connection on U. Making statements based on opinion; back them up with references or personal experience. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Since tensor products form a basis this fully defines the connection $\Sigma^\ast \nabla \otimes D$. covariant derivative electromagnetism SHARE THIS POST: will be $$\nabla_{X} T = \frac{dT}{dX} − G^{-1} (\frac{dG}{dX})T$$.Physically, the correction term is a derivative of the metric, and we’ve already seen that the derivatives of the metric (1) are the closest thing we get in general relativity to the gravitational field, and (2) are not tensors. Thanks for contributing an answer to Physics Stack Exchange! & дх” дх” ' -Tb; (Assume that the Leibnitz rule holds for covariant derivative). Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Since we have v $$\theta$$ = 0 at P, the only way to explain the nonzero and positive value of $$\partial_{\phi} v^{\theta}$$ is that we have a nonzero and negative value of $$\Gamma^{\theta}_{\phi \phi}$$. Since the path is a geodesic and the plane has constant speed, the velocity vector is simply being parallel-transported; the vector’s covariant derivative is zero. How is this octave jump achieved on electric guitar? called the covariant vector or dual vector or one-vector. This will be: $$(\Sigma^\ast \nabla\otimes D)_Z\mathfrak{t} = \bigg[(\Sigma^\ast \nabla\otimes D)_Zt^\mu_A\bigg](\Sigma^\ast \partial_\mu)\otimes d\xi^A+t^\mu_A\bigg[(\Sigma^\ast \nabla\otimes D)_Z(\Sigma^\ast \partial_\mu)\otimes d\xi^A\bigg]$$, The first term has a covariant derivative of a real-valued function. $$\Sigma^\ast (TM)=\{(\xi,v)\in W\times TM : v\in T_{\Sigma(\xi)}M\},\quad \pi(\xi,v)=\xi,$$, $$(\Sigma^\ast X)(\xi)=(\xi,X(\Sigma(\xi))).$$, $${\cal S}(\xi)={\cal S}^a(\xi)E_a(\Sigma(\xi))$$, $$S(\xi)={\cal S}^a(\xi) \Sigma^\ast E_a(\xi)$$, $$(\Sigma^\ast\nabla)_{Z}(\Sigma^\ast X)=\Sigma^\ast\bigg(\nabla_{\Sigma_\ast Z} X\bigg),\quad Z\in \Gamma(TW), X\in \Gamma(TM).$$, $${\frak t}=t^\mu_A (\Sigma^\ast \partial_\mu)\otimes d\xi^A.$$, $$\Sigma_\ast Z = \dfrac{\partial (x^\nu\circ \Sigma)}{\partial \xi^B}Z^B \partial_\nu = t^\nu_B Z^B \partial_\nu.$$. G term accounts for the change of basis derivative or ( linear ) connection on the tangent )! 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