Two definitions of a connection For me a connection $\nabla$ on a vectorbundle $E$ over a smooth manifold $M$ is a $\mathbb{R}$-bilinear map $\Gamma(TM)\times\Gamma(E)\rightarrow\Gamma(E)$ which is tensorial in the first slot and satisfies the Leibniz rule in the second. 
Now i have seen a different definition that a connection is a ${R}$-linear map $\nabla : \Gamma(E) \to \Gamma(E\otimes T^*M)$ which also satisfies some Leibniz rule.
My question is: What exactly is the bundle $\Gamma(E\otimes T^*M)$ and how are these definitions equivalent. Thanks in advance!
 A: $\Gamma(E \otimes T^*M)$ is the set of sections of the vector bundle $E \otimes T^*M = \bigcup_{p \in M} E_p \otimes T_p^*M$. The usual second definition (that I will be using) is

Definition: A connection on a vector bundle $\pi:E \to M$ is a map
          $$D: \Gamma(E) \to \Gamma(T^*M \otimes E)$$ such that for any $s_1, s_2 \in \Gamma(E)$, $D(s_1 + s_2) = Ds_1 + Ds_2$ and for any section $s \in \Gamma(E)$ and $\alpha \in C^\infty(M)$, $D(\alpha s) = d\alpha \otimes s + \alpha Ds$.

Now I will sketch one way to get between the two definitions.
($D \to \nabla$): For fixed $X \in \Gamma(TM)$, define $\nabla_X(s) := i_X(Ds)$ where, for $\omega \otimes s \in \Gamma(T^*M \otimes E)$, $i_X(\omega \otimes s) = \omega(X) s \in \Gamma(E)$. Then it's a straightforward exercise to check that this has the desired properties.
($\nabla \to D$): Let $x^1, \dots, x^n$ be local coordinates for some chart $U \subseteq M$. Define in local coordinates
    $$D(s) = \sum_{i=1}^n dx^i \otimes \nabla_{\frac{\partial}{\partial x^i}} s.$$
Then you can check that this definition has the right behaviour under a change of coordinates to define a global object. Finally, it is an easy coordinate based calculation to check the desired properties.
Finally, by a calculation in local coordinates, you can check that these constructions are inverse to each other so that the definitions are equivalent.
