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Consider following two definitions:

Definition 1. Let $M$ be a smooth manifold, let $p\in M$ and assume that $V$ is a real vector space. We will say that a $\mathbb{R}$-linear map $w:C^\infty(M)\to V$ is a tangent vector at $p$ with values in $V$ if for any $f,g\in C^\infty(M)$ $$w(fg)=f(p)w(g)+g(p)w(f).$$

And we can generalize it to the global objects.

Definition 2. Let $M$ be a smooth manifolds and assume that $E$ is smooth vector bundle over $M.$ We will say that a $\mathbb{R}$-linear map $W:C^\infty(M)\to \Gamma(E)$ is a vector field with values in $E$ if for any $f,g\in C^\infty(M)$ $$W(fg)=fW(g)+gW(f).$$

I think this is a very natural generalisation of tangent vectors and vector fields.

Question. Are there any examples of tangent vectors/vector fields with values in vector bundles which arises in the differential geometry?

Obviously except when $V=\mathbb{R}$ or $E$ is a trivial 1-dimensional vector bundle $M\times\mathbb{R}.$ I am interested if such objects appear in riemmanian, or symplectic, or poisson, or any other geometry.

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The notion of derivation is quite general, and encompasses also the definitions you have stated. One particularly fruitful example in differential geometry is the exterior derivative $$d:C^\infty(M)\to \Omega^1(M)$$ taking smooth functions to differential one-forms on the manifold. In the terms of your definition 2, the vector bundle $E$ in this case is the cotangent bundle $E=T^*M$ of the manifold.

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