The Mathematics of Soap Films: Explorations With Maple (Student Mathematical Library, Vol. 10) (Student Mathematical Library, V. 10)
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Nature tries to minimize the surface area of a soap film through the action of surface tension. The process can be understood mathematically by using differential geometry, complex analysis, and the calculus of variations. This book employs ingredients from each of these subjects to tell the mathematical story of soap films. The text is fully self-contained, bringing together a mixture of types of mathematics along with a bit of the physics that underlies the subject.The development is primarily from first principles, requiring no advanced background material from either mathematics or physics. Through the MapleR applications, the reader is given tools for creating the shapes that are being studied. Thus, you can 'see' a fluid rising up an inclined plane, create minimal surfaces from complex variables data, and investigate the 'true' shape of a balloon. Oprea also includes descriptions of experiments and photographs that let you see real soap films on wire frames. The theory of minimal surfaces is a beautiful subject, which naturally introduces the reader to fascinating, yet accessible, topics in mathematics. Oprea's presentation is rich with examples, explanations, and applications. It would make an excellent text for a senior seminar or for independent study by upper-division mathematics or science majors.
identity theorem. Also, show that f' is not continuous at 0, so real functions differ from complex analytic ones which have the property that derivatives of all orders exist and are continuous. 2.5. Gauss Curvature Although we shall primarily be concerned with mean curvature in this text, it is important to realize that yet another 'curvature' is eq'ually important. This new curvature is called the Gauss curvature and is denoted by K. Here we shall simply develop one formula for the Gauss
below, but we point out here that we need the vector fields (and curves) to be real-analytic. That is, the coordinate functions must have Taylor series which converge to the function's value at each point of the domain. In practice, everything we deal with has this property, so the reader should not fret unduly. So, let O'(t) be a curve in ]R3 and let N(t) be a vector field along 0' which has the extra property that N(t) . O"(t) = for all tEl, where O"(t) is the tangent vector to 0' at t. °
is localized to within about one atomic thickness of the interface. The inward force produces a uniform tension across the surface. To see this, let's consider an old experiment. Take a horseshoe-shaped wire with a handle and with a rod across the horseshoe which is free to slide along the horseshoe. Dip the whole apparatus in a soap solution and remove it to reveal a soap film which pulls the slider towards the handle of the horseshoe. S- HOI ","shoe Figure 3: The Force of a Soap Film Thus,
the Monge parametrization x = (t,s,x(t,s)) has Xt = (1,0,xd, Xs = (0, 1,x s )' Then, if we denote the intersection curve of the surface x with the plane by a, we have by Lemma 2.1.2 that Q = tXt+sxs = u(1, 0, Xt)+ v(O, 1, xs), where we still use the 'dot' notation for differentiation with respect to the curve's parameter. Because a is in the plane, its tangent vector is perpendicular to n: 0 = at + bs. Also, we have U . n = -aXt - bx s . Substituting b = -atl s into this equation gives since
energy is determined 1. Surface Tension 14 by the ability of the soap film to do work. In turn, the amount of work done is proportional to the change in surface area. Therefore, we have Theorem 1.4.1 (First Principle of Soap Films). A soap film takes a shape which minimizes surface area. Remark 1.4.2. These surface area minima may be local minima in the sense of calculus. Nevertheless, this property allows us to remove ourselves from the world of physical things and study soap films