X Preface the calculus of variations in what I hope is a concise and effective manner. I am grateful to my Amath 507 students for their enthusiasm and hard work and for uncovering interesting applications of the calculus of variations. I owe special thanks to William K. Smith for supervising my undergraduate thesis in the calculus of variations (35 years ago) and to Hanno Rund for teaching a fine series of courses on the calculus of variations during my graduate years. Sadly, these two wonderful teachers are now both deceased. I thank the fine editors and reviewers of the American Mathematical Society for their helpful comments. Finally, I thank my family for their encouragement and for putting up with the writing of another book. Mark Kot Chapter 1 Introduction 1.1. The brachistochrone The calculus of variations has a clear starting point. In June of 1696, John (also known as Johann or Jean) Bernoulli challenged the greatest mathematicians of the world to solve the following new problem (Bernoulli, 1696; Goldstine, 1980): Given points A and B in a vertical plane to find the path AMB down which a movable point M must, by virtue of its weight, proceed from A to B in the shortest possible time. Imagine a particle M of mass m, in a vertical gravitational field of strength g, that moves along the curve y = y(x) between the two points A = (a,ya) and B = (b,yb) (see Figure 1.1). The time of descent T of the particle is T L Lb T= fdt = j%da = j\ds = j-Jl + y'2dx, (1.1) 0 0 0 a where s is arc length, L is the length of the curve, and v is the speed of the particle. If our particle moves without friction, the law of conservation of mechanical energy guarantees that the sum of the particle's kinetic 1 2 1. Introduction 1.1. The brachistochrone 3 y A a b Figure 1.1. Curve of descent energy and potential energy remains constant. If our particle starts from rest, we may thus write 1 2 -mv + mgy = mgya . (1.2) The particle's speed is then « = VWVa - y) ■ (1.3) We now wish to find the brachistochrone (from /3paxtcrTO<;, shortest, and xpovo<;, time; John Bernoulli originally, but erroneously, wrote brachystochrone). That is, we wish to find the curve V = y(x) < ya (1.4) that minimizes the integral T-j-jfiiZ*. (1.5) vzg J V y*-y a Several famous mathematicians responded to John Bernoulli's challenge. Solutions were submitted by Gottfried Wilhelm Leibniz (1697), Isaac Newton (1695-7, 1697), John Bernoulli (1697a), James (or Jakob) Bernoulli (1697), and Guillaume l'Hôpital (1697). Leibniz provided a geometrical solution. He derived the differential equation for the brachistochrone but did not specify the resulting curve (Goldstine, 1980). Leibniz also suggested that the brachistochrone be called the tachystoptotam (from Taxt-crroq, swiftest, and twttclv, to fall). Mercifully, this suggestion was ignored. Newton's anonymous solution was published in the Philosophical Transactions; it was then reprinted in the Acta Eruditorum. Newton provided the correct answer but gave no clue to his method. Despite Newton's anonymity, John Bernoulli recognized that the work was "ex ungue Leonem" (from the claw of the Lion) and the Acta Eruditorum listed Newton in its index of authors. John Bernoulli provided two solutions. The first solution relied on an analogy between the mechanical brachistochrone and light. Bernoulli (1697a) was quite taken with Fermat's principle of least time for light and argued that the brachistochrone "is the curve that a light ray would follow on its way through a medium whose density is inversely proportional to the velocity that a heavy body acquires during its fall." He broke up the optical medium into thin horizontal layers, chose an appropriate index of refraction, and used Snelľs law of refraction and calculus to determine the shape of the brachistochrone. John Bernoulli (1718) described his second solution many years later. This second solution received little attention at the time but is now viewed as the first sufficiency proof in the calculus of variations. James Bernoulli's solution was not as elegant as that of his younger brother, but it contained the key idea of varying only one value of the solution curve at a time. This idea provided the basis for further work in the calculus of variations. James Bernoulli called his solution an oligochrone (from oAryo?, little, and xpoz^K, time). We shall see that the brachistochrone is the inverted cycloid x{4>) = a + R(

) = ya - R (1 — cos 4>), (1.6) where the parameter R is uniquely determined by the initial and terminal points. This cycloid is the curve traced by a point on the 4 1. Introduction 1.2. The terrestrial brachistochrone 5 Figure 1.2. Cycloid for R = ~ya and a = 0 circumference of a circle of radius R rolling along the bottom of the horizontal line y = ya (see Figure 1.2). Huygens (1673, 1986) had previously shown that an inverted cycloid is a tautochrone (from ravro or to avro, the same, and xpoijo^; time): the time for a heavy particle to fall to the bottom of this curve is independent of the upper starting point. To John Bernoulli's astonishment, the brachistochrone was Huygens' tautochrone. The brachistochrone is one of many problems where we wish to determine a function, y(x), that minimizes or maximizes the integral b J[y(x)} = Jf(x,y(x),y'(x))dx. (1.7) a Leonhard Euler first devised a systematic method for solving such problems. In the remainder of this chapter, we will examine three other problems that involve minimizing or maximizing integrals. We will first look at another brachistochrone problem, for travel through the earth. We will then look at the problem of finding the shortest path between two points on some general surface. Finally, we will look at the "soap-film problem," the problem of minimizing the surface area of a surface of revolution. All of these problems can be attacked using the calculus of variations. 1.2. The terrestrial brachistochrone History repeats itself. In August of 1965, Scientific American published an article on "High-Speed Tube Transportation" (Edwards, 1965). Edwards proposed tube trains that would fall through the earth, pulled by gravity and helped along by pneumatic propulsion. The advantages cited by Edwards included: (1) It brings most of the tunnel down into deep bedrock, where the cost of tunneling — by blasting or by boring — is reduced and incidental earth shifts are minimized; the rock is more homogeneous in consistency and there is less likelihood of water inflow. (2) The nuisance to property owners decreases with depth, so the cost of easements should be lower. (3) A deep tunnel does not interfere with subways, building foundations, utilities, or water wells.. .. (4) The pendulum ride is uniquely comfortable for the passenger---- Lest you think this pure fantasy, a pneumatic train was constructed in New York City, under Broadway, from Warren Street to Murray Street, in 1870 by Alfred Ely Beach (an early owner of Scientific American). This was New York City's first subway (Roess and Sansone, 2013). You can see a drawing of the pneumatic train on the wallpaper in older Subway Sandwich shops. Cooper (1966a) then pointed out that straight-line chords lead to needlessly long trips through the earth. He used the calculus of variations to derive a differential equation for the fastest tunnels through the earth and integrated this equation numerically. Venezian (1966), Mallett (1966), Laslett (1966), and Patel (1967) then found first integrals and analytic solutions for this problem. See Cooper (1966b) for a summary. Let us take a closer look at this terrestrial brachistochrone problem. Assume that the earth is a homogeneous sphere of radius R. Consider a section through the earth with polar coordinates centered at the heart of the earth (see Figure 1.3). Imagine a particle of mass 6 1. Introduction 1.3. Geodesies 7 where g is the magnitude of the gravitational acceleration at the surface of the earth. For a particle starting at rest at the surface of the earth, conser vation of energy now implies that 1,1 mg 2 1 2mV +2lT = 2m9R so that v = (1.10) (1.11) It follows that the total travel time is T - I d9. (1.12) Figure 1.3. Path through the earth m that moves between two points, A = {ra,0a) and B = (rb, 9b), on or near the surface of the earth. We now wish to find the planar curve 7 that minimizes the travel time T T = Idt = J fsdS = j\ds = Jl ^dr2 + r2dB2 (1-8) 0 7 7 7 between A and B, where s is arc length and v is the speed of the particle. When a particle is outside a uniform spherical shell, the shell exerts a gravitational force equal to that of an identical point mass at the center of the shell. A particle inside the shell feels no force (see Exercise 1.6.3). By integrating over spherical shells of different radii (Exercise 1.6.4), one can show that the gravitational potential energy within a spherical and homogeneous earth can be written We will look at this problem in greater detail later. We shall see that the terrestrial brachistochrone is a hypocycloid, the curve traced by a point on the circumference of a circle of radius either [R - (Sab/v)} (see Figure 1.4) or of radius Sab/k (see Figure 1.5), where Sab is the arc length along the surface of the earth between A and B, as it rolls inside a circle of radius R. The fastest Amtrak train makes the 400 mile trip between Boston and Washington, D.C., in six and a half hours. A tube train moving along a straight-line chord between Boston and Washington would penetrate 5 miles into the earth and take 42 minutes. The fastest tube train along a hypocycloid would, in turn, penetrate 125 miles into the earth and take 10.7 minutes. 1.3. Geodesies I do not want to give the impression that the calculus of variations is only brachistochrones. In this and the next section, we will look at two other classic problems. A line is the shortest path between two points in a plane. We also wish to find shortest paths between pairs of points on other. „„„„^1 o.n.(o™ Tr> finrl +Vidof> o-pnrlpsirs. we must minimize 8 1. Introduction 1.3. Geodesies 9 The simplest case arises when the surface is a level set for one of the coordinates in a system of orthogonal curvilinear coordinates. The arc length can then be written using the scale factors of the coordinate system. Consider, for example, two points, A and B, on a sphere of radius R centered at the origin. We wish to join A and B by the shortest, continuously differentiable curve lying on the sphere. We start by specifying position, r{x,y,z) = xi + yj + zk, (1.13) using the Cartesian coordinates x, y, and z and Cartesian basis vectors i, j, and k. For points on the surface of a sphere, we now switch to the spherical coordinates r, 9, and (see Figure 1.6). Since the position vector r now takes the form r(r,6,4>) = r sin 9 cos i + r sin 6 sin 4> j + r cos 9 k. (1.15) Since this position vector depends on r, 9, and (j), *-S*+S-+S*- (116) The three partial derivatives on the right-hand side of this equation are vectors tangent to motions in the r, 9, and

directions and ^-lUrsinfl (1.18) d\\ hr = dr dr liar = 1, he-Wae = r, h2 . For a sphere of radius r = R, this element reduces to ds = R^Jd92 + sin2 9 dc)>2 . (1.20) If we assume that = (f>(9), finding the curve that minimizes the arc length between the points A = (9a, a) and B — (9b, (j>b) simplifies to finding the function {9) that minimizes the integral ds = Rj yfi+ sin2 q (fy/My d6 (1 21) subject to the boundary conditions {0a) = (9b) = 4>b. (122) We will see, later, that the shortest paths on a sphere are arcs of great circles. Unfortunately, we cannot expect every interesting surface to be the level set for some common coordinate. We may, however, hope to represent our surface parametrically. We may prescribe the x, y, and z coordinates of points on the surface using the parameters u and v and write our surface in the vector form r(u, v) = x(u,v) i + y(u,v) j + z(u, v) k. ;i.23) We can now specify a curve on this surface by prescribing u and v in terms of a single parameter — call it t — so that u = u(t), v — v(t). The vector dr dt dr . dr . ou av (1.24) (1.25) is tangent to both the curve and the surface. We find the square of the distance between two points on a curve by integrating ds2 dr ■ dr dr , dr , — du+—dv ou av dr , dr -w~du+ —dv ou dv along the curve. Equation (1.26) is often written ds2 = Edu2 +2F dudv + Gdv2, where „ dr dr „ E = — ■ — , F dr dr dr dr du dv ' dv dv (1.26) (1.27) (1.28) The right-hand side of equation (1-27) is called the first fundamental form of the surface. The coefficients E(u,v), F(u,v), and G(u, v) have many names. They are sometimes called first-order fundamental magnitudes or quantities. Other times, they are simply called the coefficients of the first fundamental form. The distance between the two points A = (ua,va) and B = (ub, vb) on the curve u = u(t), v = v(t) may now be written tb -If® „ ^du dv _ + 2FltTt+G dt, (1.29) i. introduction jL.a. vjeouesica with u{ta) = ua , v{ta) u(tb) - ub , v(tb) = vh . (1.30) In this formulation, we have two dependent variables, u(t) and v(t), and one independent variable, t. If v can be written as a function of u, v — v(u), we can instead rewrite our integral as du J \du) (1.31) with v{ua) = va, v(ub) = vb. (1.32) This is now a problem with one dependent variable and one independent variable. To make all this concrete, let us take, as an example, the pseu-dosphere (see Figure 1.7), half of the surface of revolution generated by rotating a tractrix about its asymptote. If the asymptote is the 2-axis, we can write the equation for a pseudosphere, parametrically, as Since and r(u, v) = o sin u cos v i + a sin u sin v j + a ^cos u + In tan k . dr du (1.33) (1.34) (a cos u cos v, a cos u sin v, —a sin u + a esc u) = — = (—a sin u sm v, a sin « cos v, 0). an the first-order fundamental quantities reduce to E = vu ■ vu = a2 cot2 u, F = ru • r„ = 0, G = r„ • rv = a2 sin2 u. (1.35) (1.36) (1.37) (1.38) Figure 1.7. Pseudosphere To determine a geodesic on the pseudosphere, we must thus find a curve, u = u{€) and v = v(t), that minimizes the arc-length integral tb _ s = a j Vcot2 u u2 + sin2 u v2 dt (1.39) subject to the boundary conditions U(ta) = Ua , V{ta) = Va , u(tb) = Ub , v{tb) = Vb . (1.40) Alternatively, we may look for a curve, v = v(u), that minimizes the integral (1.41) (1.42) ;ot2 u + sin2 u ( ^ 1 1 du. subject to the boundary conditions v{ua) = va, v(ub) = vb. For other examples, see Exercise 1.6.6. x. introduction 1.4. Minimal surfaces it) John Bernoulli (1697b) posed the problem of finding geodesies on convex surfaces. In 1698, he remarked, in a letter to Leibniz, that geodesies always have osculating planes that cut the surface at right angles. (An osculating plane is the plane that passes through three nearby points on a curve as two of these points approach the third point.) This geometric property is frequently used as the definition of a geodesic curve, irrespective of whether the curve actually minimizes arc length. Later, Euler (1732) derived differential equations for geodesies on surfaces using the calculus of variations. This was Euler's earliest known use of the calculus of variations. Finding shortest paths is easiest on simple surfaces of revolution. Geodesies on surfaces of revolution satisfy a simple first integral or "conservation law" that was first published by Clairaut (1733). Jacobi (1839), in a tour de force, succeeded in integrating the equations of geodesies for a more complicated surface, a triaxial ellipsoid. 1.4. Minimal surfaces We may minimize areas as well as lengths. Consider two points, y(a) = ya , y(b) = yb, (1.43) in the plane (see Figure 1.8). We wish to join these two points by a continuously differentiable curve, V = y(x) > 0. (1.44) in such a way that the surface of revolution, generated by rotating this curve about the i-axis, has the smallest possible area S. In other words, we wish to minimize & S = 2n J y(x) yjl + y'2 dx . (1.45) Some of you will recognize this as the "soap-film problem." Suppose we wish to find the shape of a soap film that connects two wire hoops. For a soap film with constant film tension, the surface energy is proportional to the area of the film. Minimizing the surface energy of the film is thus equivalent to minimizing its surface area (Isenberg, Figure 1.8. Profile curve 1992; Oprea, 2000). (For a closed soap bubble, without fixed boundaries, excess air pressure within the bubble prevents the surface area of the bubble from shrinking to zero.) Euler (1744) discovered that the adenoid, the surface generated by a catenary or hanging chain (see Figure 1.9), minimizes surface area. As you doubtless know, however, from playing with soap films, if you pull two parallel hoops too far apart, the catenoid breaks, leaving soap film on the hoops. This was first shown analytically by Goldschmidt (1831). For two parallel, coaxial hoops of radius r, the area of a catenoid is an absolute minimum if the distance between the hoops is less than 1.056 r. This area is a relative minimum for distances between 1.056 r and 1.325 r. For distances greater than 1.325 r, the catenoid breaks and the solution jumps to the discontinuous Goldschmidt solution (two disks). Joseph Lagrange (1762) then proposed the general problem of finding a surface, z = f(x, y), with a closed curve C as its boundary, that has the smallest area. That is, we now wish to minimize a double 1.4. Minimal surfaces 17 Figure 1.9. Catenoid integral of the form 5 = // yfi+ß + fidxäy (1.46) (see Exercise 1.6.7), where dfi is the projection of the closed curve C onto the (x, y) plane and $7 is the interior of this projection. This problem has been known, starting with Lebesgue (1902), as Plateau's problem, in honor of Joseph Plateau's extensive experiments (Plateau, 1873) with soap films. Lagrange showed that a surface that minimizes integral (1-46) must satisfy the minimal surface equation (1 + &f**-2fxfyfxy + (l + fx)fyy = 0 inear, elliptic, seco: mstraints on the fi different minimal surfaces. (1.47) a quasilinear, elliptic, second-order, partial differential equation Dif-^^!ramtf °\the fanCti0n (e-g., Exercise 1.6.10) yield Figure 1.10. Helicoid Jean-Baptiste-Marie-Charles Meusnier (1785) soon gave equation (1.47) a geometric interpretation. At each point P of a smooth surface, choose a vector normal to the surface, cut the surface with normal planes (that contain the normal vector but that differ in orientation), and obtain a series of plane curves. For each plane curve, determine the curvature at P. Find the minimum and maximum curvatures (from amongst all the plane curves passing through P). These are your principal curvatures. Meusnier showed that the minimal surface equation implies that the mean curvature (the average of the principal curvatures) is zero at every point of the minimizing surface. As a result, any surface with zero mean curvature is typically referred to as a minimal surface, even if it does not provide an absolute or relative minimum for surface area. Meusnier also discovered that the catenoid and the helicoid, the surface formed by line segments perpendicular to the axis of a circular helix as they go through the helix (see Figure 1.10), satisfy Lagrange's x. i hi i iiuuci ion minimal surface equation. (Meusnier, like Lagrange, seemed unaware of Euler's earlier analysis of the catenoid.) The study of minimal surfaces has grown to become one of the richest areas of mathematical research. In the remainder of this book, we will look at many other problems in the calculus of variations. 1.5. Recommended reading Goldstine (1980), Fraser (2003), Kolmogorov and Yushkevich (1998), and Kline (1972) provide useful historical surveys of the calculus of variations. Icaza Herrera (1994), Sussmann and Willems (1997), and Stein and Weichmann (2003) have written stimulating historical articles about the brachistochrone problem. An experimental study of the brachistochrone (using a "Hot Wheels" car) was carried out by Phelps et al. (1982). The original 1697 solutions of John and Jacob Bernoulli can be found, translated into English, in Struik (1969). John Bernoulli's solution was recently reviewed by Erlichson (1999) and reviewed and generalized by Filobello-Nino et al. (2013). If the endpoints A and B lie above the surface of the earth, but at vastly different heights, the gravitational field is no longer constant. One must instead determine the curve of swiftest descent in an attractive, inverse-square, gravitational field. This problem has been discovered repeatedly. Recent treatments include those of Singh and Kumar (1988), Parnovsky (1998), Tee (1999), and Hurtado (2000). Goldstein and Bender (1986) analyzed the brachistochrone in the presence of relativistic effects and Farina (1987) showed that John Bernoulli's optical method can also be used to solve this relativistic problem. Kamath (1992) determined the relativistic tautochrone using fractional calculus. The idea of high-speed tunnels through the earth is quite old. In Lewis Carroll's (1894) Sylvie and Bruno Concluded, Mein Herr describes a system of railway trains, without engines, powered by gravity: i.o. itecommenaea reading iy "Each railway is in a long tunnel, perfectly straight: so of course the middle of it is nearer the centre of the globe than the two ends: so every train runs half-way down-hill, and that gives it force enough to run the other half up-hill." To which a protagonist replies: "Thank you. I understand that perfectly," said Lady Muriel. "But the velocity, in the middle of the tunnel, must be something fearful!" You can also find a homework problem, about a tunnel-train between Minneapolis and Chicago, in Brooke and Wilcox (1929). See also Kirmser (1966). Edwards' (1965) article reignited keen interest in gravity-powered transportation and inspired the articles by Cooper (1966a,b), Venezian (1966), Mallett (1966), Laslett (1966), and Patel (1967) on the terrestrial brachistochrone. Aravind (1981) applied John Bernoulli's optical method to the terrestrial brachistochrone and Prussing (1976), Chan-der (1977), McKinley (1979), and Denman (1985) pointed out that terrestrial brachistochrones are also tautochrones. Stalford and Garrett (1994) analyzed the terrestrial brachistochrone using differential geometry and optimal control theory. Struik (1933), Caratheodory (1937), and Kline (1972) summarize the early history of the study of geodesies. Geodesies are an important topic in differential geometry (Struik, 1961; Oprea, 2007), Riemann-ian geometry (Berger, 2003), and geometric modeling (Patrikalakis and Maekawa, 2002). See Bliss (1902) for examples of geodesies on a toroidal anchor ring and Sneyd and Peskin (1990) for examples of geodesic trajectories on general tubular surfaces. Isenberg (1992) and Oprea (2000) provide interesting and readable introductions to the science and mathematics of soap films. Barbosa and Colares (1986), Nitsche (1989), Fomenko (1990), and Fomenko and Tuzhilin (1991) do an excellent job of presenting the history and theory of minimal surfaces. _. i.o. iiixercises 1.6. Exercises 1.6.1. Descent time down a cycloidal curve. Show that the descent time down the cycloidal curve x() = a + R() = ya - R(l - cos i). 1.6.4. Potential energy inside the earth. Use your results from the last problem and integrate over shells of appropriate radii to show that the potential energy of a point mass m in a spherical and homogeneous earth can be written, to within an additive constant, as V(r) = j-g.r , (1.53) where R is the radius of the earth, g is the magnitude of the gravitational acceleration at the surface of the earth, r is the distance of the point mass from the center of the earth, and p is the (volumetric) density of the earth. Figure 1.12. Geometry of a spherical shell 1.6.5. Gauss's law. Gauss's flux theorem for gravity states that the gravitational flux through a closed surface is proportional to the enclosed mass. Gauss's theorem can be written in differential form, using the divergence theorem, as V-g = -4nGp, (1.54) where G is the universal gravitational constant, p is the (volumetric) density of the enclosed mass, g = F/m is the gravitational field intensity, m is the mass of a test point, and F is the force on this test mass. (a) Use this theorem to determine the force F(r) acting on mass m at point P due to the gravitational attraction of a thin homogeneous spherical shell of mass M, surface density a, and radius x. Assume that point P is a distance r from the center of the shell. Be sure to consider the case where point P is inside the shell (r < x) as well as outside the shell (r > x). (b) Assume that F(r) = —dV/dr, where V(r) is the gravitational potential energy. Integrate the above force (starting at a reference point at infinity) to rederive the potential energy in Exercise 1.6.1. (c) Use Gauss's flux theorem to determine the force F(r) acting on mass rn at point P due to the gravitational attraction of a uniform solid sphere of mass M, density p, and radius R. Be sure to consider the case where point P is inside the shell (r < R) as well as outside the shell (r > R). (d) Integrate the above force (starting at a reference point at infinity) to rederive the potential energy in Exercise 1.6.2. 1.6.6. First fundamental forms. Determine the first fundamental form for three of the following seven surfaces. The surfaces you may choose from are: (a) the helicoid x = u cos v , y = u sin v . z = av; (1.55) (b) the torus x = (b + a cos u) cos v , x — (b + acosu) cosv , z = asinu; (1.56) (c) the catenoid x = a cosh — cos v , y = a cosh — sin v , z = u ; a a (d) the general surface of revolution x = f(u) cos v, y = f(u) sin v, z — g(u); (e) the sphere (with alternate parameterization) Aa2u 4a2v x - 4a2 + u2 + v2 ' 4a2 + v? + v2 ' (1.57) (1.58) (1.59) z = a 4a2 - u2 - v2 4a2 + u2 + v2 ' (f) the ellipsoid x = a cos u cos v . y = b cos u sin v , z — c sin u: (g) the hyperbolic paraboloid x = a(u + v), y = b(u-v). z = uv. (1.60) (1.61) 1.6.7. Surface area. Consider a surface written in the vector form r(u,v) = x(u, v) i + y{u,v) j + z(u, v) k, (1-62) where u and v are parameters. (a) Justify or motivate the surface-area formula S = jj ||r« x r„|| dudv. (1.63) (b) Show that the above surface-area formula can also be written as S = ?-F2dudw, (1.64) where £, F, and G are the coefficients of the first fundamental form. (c) Write the surface z = f(x,y) (1.65) in vector form and show that the above formulas for area imply that S = If ^JÍ+fT+fž dxdy. (1.66) n 1.6.8. Surface area of a hyperbolic paraboloid. Consider the hyperbolic paraboloid r(u, v) = ui + vj + uvk. (1-67) Determine the surface area for that portion of the paraboloid that is specified by values of u and v that lie in the first quadrant of (u, v) parameter space between the positive u- and u-axes and the circle w2 + v2 1 (1.68) 1.6.9. Surface area of a helicoid. Find the area of the portion of the helicoid r(u, v) = u cos v i + u sin v j + bv k that is specified by 0 < u < a and 0 < v < 2-k. (1.69) Figure 1.13. Scherk's first minima] surface 1.6.10. Scherk's minimal surface. Take the minimal surface equation, equation (1-47), and look for a solution of the form f(x,y) = g(x) + h(y). (1.70) Show that the resulting differential equation is separable. Solve for g(x) and h(y) to obtain Scherk's (first) minimal surface, cos (x/c)~ cos (y/c) This surface was the first minimal surface discovered after the catenoid and the helicoid. A piece of this surface, for c — 1, — it/2 < x < tt/2, and -7t/2 < y < tt/2, is shown in Figure 1.13. f{x,y) = cln (1.71)