  It is well known that an $m$-dimensional Riemannian manifold can be locally
isometrically embedded into the $m+1$-dimensional Euclidean space if and only
if there exists a symmetric 2-tensor field satisfying the Gauss and Codazzi
equations. In this paper, we prove that two known intrinsic conditions, which
were obtained previously by Weiss, Thomas and Rivertz, are sufficient to ensure
the existence of such symmetric 2-tensor field under certain generic condition
when $m=3$. Note that, in the case $m=3$, a symmetric 2-tensor field satisfying
the Gauss equation does not satisfy the Codazzi equation automatically, which
is different from the cases $m \geq 4$. In our proof, the symbolic method,
which is a famous tool known in classical invariant theory, plays an important
role. As applications of our result, we consider $3$-dimensional warped product
Riemannian manifolds whether they can be locally isometrically embedded into
$\mathbb{R}^4$. In some case, the Monge-Amp\`{e}re equation naturally appears.
