A (covariant)$(0,2)$-tensor can be written as:

$$T = \sum_{\mu}\sum_{\nu}t_{\mu \nu} e^{\mu}\bar{\otimes}e^{\nu}\tag{1}$$

with the particular basis vectors $e^{\mu}\bar{\otimes}e^{\nu}$ that spans the $\mathfrak{V}^{*}\otimes \mathfrak{V}^{*}$, the Tensor Product vector space.

If the one wants to compute the tensor components of $(1)$, with respect to another basis,then the change of basis occurs like:

$$ t'_{\mu\nu} = M^{\gamma}_{\mu '}M^{\delta}_{\nu '}t_{\gamma\delta} \tag{2}$$ $$***$$

Now, with manifold theory (and therefore a further construct called Fibre Bundle, which I do not defined here) the object $(1)$ becomes a tensor field and the components $(2)$ becomes also, functions of the position:

$$T(x^{\ell}) = \Big(\sum_{\mu}\sum_{\nu}t_{\mu \nu} e^{\mu}\bar{\otimes}e^{\nu}\Big)(x^{\ell})\tag{3}$$

and therefore,

$$ t'_{\mu\nu}(x^{\ell}) = \Big(M^{\gamma}_{\mu '}M^{\delta}_{\nu '}t_{\gamma\delta}\Big)(x^{\ell}) \tag{4}$$

A tensor field,roughly speaking, is a object that tells you that on each point of the manifold there is a tensor (this intuitive notion is analogous of a vector field). The following discussion will be based upon the specific notion of a tensor given br $(4)$.

Well, there is a branch of mathematics (a collection of tools a I would say) that merge Differential geometry and multilinear algebra into Tensor Calculus and my doubt is about that. In Tensor calculus for physics and engeneering there is a lot of theory about differential part of tensors [as $(4)$].

To differentiate $(4)$ we need the notion of covariant derivative:

$$ \nabla _{\delta}T_{\mu \nu} := \partial_{\delta}T_{\mu \nu} - \Gamma^{\gamma}_{\mu \nu} T_{\gamma \nu} - \Gamma^{\xi}_{\mu \nu}T_{\mu \xi}$$

And with that, we can construct another concept of derivation operation is called Lie derivative:

$$\mathcal{L}_{\xi} T_{\mu \nu} = \xi^{\gamma}\nabla _{\gamma}T_{\mu \nu} +T_{\mu \ell}\nabla_{\ell} \xi^{\ell} +T_{\mu \delta}\nabla \xi^{\delta}$$

Also, the Riemman tensor (the curvature tensor) are defined with respect of Christoffell symbols, the torsion tensor the same....

But what about the integral part of Tensor "Calculus"? I mean, what suppose to mean these kind of integral operations:

$$ \int T_{\mu \nu}$$

$$ \int T_{\mu \nu}l^{\mu}l^{\nu}$$

I really appreciate a basic,but complete and serious, explanation and some introductory references of the subject(with good calculation examples and friendly discussions).

  • $\begingroup$ The indices in the second and third terms of your covariant derivative and your Lie derivative are wrong. In fact, they don’t make sense. $\endgroup$ – G. Smith Jan 11 at 22:23
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    $\begingroup$ When you ask "what about the ..." are you getting these integrals from a text? If so which one? Or are you guessing at what a tensor integration would look like? In general integration is done on differential forms. $\endgroup$ – ggcg Jan 11 at 22:30

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