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](http://sipi.usc.edu/%7Eajoshi/mathesis.pdf)* the covariant derivation have been considered as a map $\Gamma(TM^\*\otimes f^{-1}TN)\to \Gamma(TM^∗ \otimes TM^∗\otimes f^{-1}TN)$.



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> why this two consideration is equivalent?
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Question on Harmonic maps between Riemannian manifolds
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>


