  The explicit construction of direct and inverse Fourier's vector transform
with discontinuous coefficients is presented. The technique of applying
Fourier's vector transform with discontinuous coefficients for solving problems
of mathematical physics.Multidimensional integral transformations with
non-separated variables for problems with discontinuous coefficients are
constructed in this work. The coefficient discontinuities focused on the of
parallel hyperplanes. In this work explicit formulas for the kernels in the
case of ideal coupling conditions are obtained; the basic identity of the
integral transform is proved; technique of integral transforms is developed
