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In Theory of harmonic maps, main goal is to find minimum of Dirichlet energy function which is defined as follows: $$E(f):=\frac{1}{2}\int_M\|df\|^2dvol_g\qquad f:(M,g)\to(N,h).$$

In many Books such as Calculus of Variations and Harmonic Maps-Hajime Urakawa, used the covariant derivation as a map $\Gamma(f^{-1}TN)\to \Gamma(f^{-1}TN)$ and in another Reference such as Graduate thesise of A.A. joshi the covariant derivation have been considered as a map $\Gamma(TM^*\otimes f^{-1}TN)\to \Gamma(TM^∗ \otimes TM^∗\otimes f^{-1}TN)$.

why this two consideration is equivalent?

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    $\begingroup$ Can you copy exactly the passages that define the covariant derivation as a map $\Gamma(f^{-1}TN)$ to itself? Is that supposed the be the action of $\nabla_X$ for some vector field $X$? $\endgroup$
    – Willie Wong
    Commented May 23, 2016 at 18:47
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    $\begingroup$ By the way, your question has nothing to do with calculus of variations, nor on the precise form of the energy functional. You can pretty much delete everything before "In many Books such as ..." // Your question is really about pullback bundles and their pullback connections. $\endgroup$
    – Willie Wong
    Commented May 23, 2016 at 18:49
  • $\begingroup$ @Willie Wong. Yes. that is supposed the be the action of $\nabla_X$ for $X\in\mathfrak{X}(M)$. $\endgroup$
    – C.F.G
    Commented May 23, 2016 at 18:56
  • $\begingroup$ And in Joshi the object being considered is the action of $\nabla$ itself, without fixing a vector field. They are not the same thing. (This explains the first $TM^*$ in Joshi's RHS.) Other than that, the two are largely the same. The fact that Joshi still has an extra factor of $TM^*$ is because when studying harmonic maps the objects you study are always sections of $TM^* \otimes f^* TN$ and not just $f^*TN$. $\endgroup$
    – Willie Wong
    Commented May 23, 2016 at 19:02
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    $\begingroup$ You should post your second question separately. $\endgroup$
    – Deane Yang
    Commented May 24, 2016 at 12:26

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the reason is the directional of derivatives allow us to multiply a tensor field and then we can introduce orthogonal coordinate by the free tensors and use the harmonic maps of covariant derivation (the levi-civita connection type) to study manifolds (in differential geometry)!

see:

https://en.wikipedia.org/wiki/Covariant_derivative

and

http://sipi.usc.edu/~ajoshi/mathesis.pdf

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