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However, in many questions of physics and mechanics there appear important functionals defined on multidimensional objects and surfaces, for example on the space of two-dimensional surfaces with a fixed boundary. An important example is the area functional, which associates an area with each such surface. Another example, closely connected with the previous one, is the Dirichlet functional. The connection between these functionals is in many 21 22 I. THE CLASSICAL THEORY OF MINIMAL SURFACES respects analogous to the well-known connection between the length func- tional and the action functional.
There are many mathematical trends that arise from specific physical and applied problems, but by no means all of them, like Plateau's problem, are so closely associated with such a wealth of various mathematical theories. Within the bounds of the theory of minimal surfaces there are interlaced such modern theories as differential equations, Lie groups, and algebras, homology and cohomology, bordisms, and so on. Let us illustrate by simple examples some basic concepts, techniques and results worked out in various periods of development of Plateau's problem.
These forces of mutual attraction cause the appearance of surface tension on the interface between the media. In contrast to water molecules, soap molecules are formed by long thin apolar hydrocarbon chains with a polar oxygen group on one end of the chain. A soap molecule is stretched: one end of it (the polar end) is COO Na+ , and the other end (the apolar end) consists of 12-18 carbon atoms. The polar end is soluble in water (hydrophilic). while the apolar end, by contrast, is repelled by water (hydrophobic).