This chapter describes the fourth of Liouville’s theorems, which concerns volume derivatives on deforming domains. This transformation theory provides a mathematical formulation of several physical phenomena involving material transport. Its applications, however, spread, beyond mathematical physics, into various areas in analysis and engineering. The first section is a summary. In the second section, we confirm a fundamental elliptic \(H^1\) regularity on Lipschitz domains, and then turn to \(H^2\) regularity on convex domains in accordance with the second fundamental form. In the third section, we describe the above mentioned Liouville’s theorem and application to the free boundary problem. In the fourth section we turn to the problem of fluid mechanics, associated with both coordinates of Lagrange and Euler, to reach the blowup of the solution to irrotational Euler flow. In the fifth section, we describe an abstract theory of shape optimization. The final section is devoted to Hadamard’s variational formulae for Green’s function of Laplacians. The first volume derivative is used for modeling of fundamental equations in physics. It is to be a basic tool to approach the principle of self-organization in the following chapter, besides the scaling described in the previous chapter. The second volume derivative, on the other hand, is more associated with geometric objects, the second fundamental form of the boundary.