  Let $X$ be a smooth complete intersection over $\mathbb{C}$ of dimension
$n-k$ in the projective space $\mathbf{P}^n_{\mathbb{C}}$, for given positive
integers $n$ and $k$. For a given integral homology cycle $[\gamma] \in
H_{n-k}(X(\mathbb{C}),\mathbb{Z})$, the period integral is defined to be a
linear map from the de Rham cohomology group to $\mathbb{C}$ given by $[\omega]
\mapsto \int_\gamma \omega$. The goal of this article is to interpret this
period integral as a linear map from the polynomial ring with $n+k+1$ variables
to $\mathbb{C}$ and use this interpretation to develop a deformation theory of
period integrals of $X$. The period matrix is an invariant defined by the
period integrals of the \textit{rational} de Rham cohomology, which compares
the \textit{rational} structures ($\mathbb{Q}$-subspace structures) of the de
Rham cohomology over $\mathbb{C}$ and the singular homology with coefficient
$\mathbb{C}$. As a main result, when $X'$ is another projective smooth complete
intersection variety deformed from $X$, we provide an explicit formula for the
period matrix of $X'$ in terms of the period matrix of $X$ and the Bell
polynomials evaluated at the deformation data. Our result can be thought of as
a modern deformation theoretic treatment of the period integrals based on the
Maurer-Cartan equation of a dgla (differential graded Lie algebra).
