Let $u(t)$ and $v(t)$ be functions in $C^{\infty}$. Then let $A(u)$ be an operator.
A valid reference mentioned that the second-order Taylor expansion of the operator $A$ is:
$$A(u+v) = A(u) + dA(u)[v] + \int_{0}^{1}(1-\alpha)d^{(2)}A(u + \alpha v)[v]^{2} d\alpha$$
What does the notation $dA(u)[v]$ and $d^{(2)}A(u + \alpha v)[v]^{2}$ mean?