This paper concerns with surface integrals in a real locally convex spaces \(Z\).
The first part is generally theoretical; here surface measures on topological surfaces are constructed and fundamental properties of these measures are established. More precisely, let \(Z_0\) be a normed space which is continuously imbedded in \(Z\) and \(\mu\) a signed Radon measure differentiable along the direction in \(Z_0\). Suppose that \(X\) is a closed hypersubspace of \(Z\) which does not contain \(Z_0\), \(a\in Z_0\setminus X\), \(R_a\) is the one-dimensional subspace generated by \(a\), \(A\) is an open set in \(X\) and \(f: A\to R_a\) is a continuous function having only continuous bounded derivative \(f': A\to (X\cap Z_0)\). The graph of \(f\) is denoted by \(G= G(X,A,a,f)\). Roughly speaking, the surfaces considered in this issue are infinite-dimensional manifolds that glue \(G(X,A,a,f)\) together as local pieces. Having established an invariance theorem which asserts that elements of surface measures \(\mu_G\) on \(G\) (the precise definition is omitted here) does not depend on a particular expression of \(G= G(X,A,a,f)\) and coincides with each other on the common domain of the different graphs, a surface measure is introduced.
The second part is devoted to the applications of the above arguments. The formula of iterated integration, the formula of integration by parts and the Gauss-Ostrogradskii and Green formulas are developed.