  In this paper, by taking into account the beginning of the hypersurface
theory in Euclidean space $E^4$, a practical method for the matrix of the
Weingarten map (or the shape operator) of an oriented hypersurface $M^3$ in
$E^4$ is obtained. By taking this efficient method, it is possible to study of
the hypersurface theory in $E^4$ which is analog the surface theory in $E^3$.
Furthermore, the Gaussian curvature, mean curvature, fundamental forms and
Dupin indicatrix of $M^3$ is introduced.
