  We provide a rigorous numerical computation method to validate tubular
neighborhoods of normally hyperbolic slow manifolds with the explicit radii for
the fast-slow system \begin{equation*} \begin{cases} x' = f(x,y,\epsilon), and
y' =\epsilon g(x,y,\epsilon). & \end{cases} \end{equation*} Our main focus is
the validation of the continuous family of eigenpairs $\{\lambda_i(y;\epsilon),
u_i(y;\epsilon)\}_{i=1}^n$ of $f_x(h_\epsilon(y),y,\epsilon)$ over the slow
manifold $S_\epsilon = \{x = h_\epsilon(y)\}$ admitting the graph
representation. In order to obtain such a family, we apply the interval
Newton-like method with rigorous numerics. The validated family of eigenvectors
generates a vector bundle over $S_\epsilon$ determining normally hyperbolic
eigendirections rigorously. The generated vector bundle enables us to construct
a tubular neighborhood centered at slow manifolds with explicit radii.
Combining rate conditions for providing smoothness of center-(un)stable
manifolds, we can validate smooth tubular neighborhoods with diffeomorphic
family of affine change of coordinates, as well as several extensions such as
conic and star-shaped neighborhoods. Our procedure provides a systematic
construction of smooth neighborhoods of slow manifolds in an explicit range
$[0,\epsilon_0]$ of $\epsilon$ with rigorous numerics.
