By Mariano Giaquinta, Guiseppe Modica, Jiri Soucek

This monograph (in volumes) offers with non scalar variational difficulties coming up in geometry, as harmonic mappings among Riemannian manifolds and minimum graphs, and in physics, as good equilibrium configuations in nonlinear elasticity or for liquid crystals. The presentation is selfcontained and available to non experts. subject matters are taken care of so far as attainable in an ordinary means, illustrating effects with easy examples; in precept, chapters or even sections are readable independently of the final context, in order that elements will be simply used for graduate classes. Open questions are usually pointed out and the ultimate component of each one bankruptcy discusses references to the literature and occasionally supplementary effects. eventually, an in depth desk of Contents and an intensive Index are of aid to refer to this monograph

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**Example text**

A point P = (x, y) is at a distance x 2 + (y − k)2 from F and y + k from f . The equation of the curve is thus given by x 2 + (y − k)2 = y + k that is, squaring both sides x 2 + (y − k)2 = (y + k)2 . 13 The Parabola 35 Fig. 26 This reduces to x 2 − 2yk = 2yk that is y= x2 . 4k Conversely, given an equation as in the statement, the conclusion follows at once 1 1 by forcing a = 4k , that is by choosing k = 4a . This time, we notice that the origin (0, 0) is a point of the parabola with equation y = ax 2 .

4. Determine the type of the conic Γ . Determine its “metric elements” with respect to the basis R (length of the axis, position of the foci, eccentricity). 12 In a rectangular system of coordinates in solid space, consider the cone with equation x 2 + 2y 2 − 3z2 = 0. Determine all the planes whose intersection with the cone is a circle. 13 In a rectangular system of coordinates in solid space and for strictly positive numbers a, b, c, prove that the quadric abz = cxy is a hyperbolic paraboloid having two lines in common with the hyperboloid of one sheet x 2 y 2 z2 − + = 1.

Ax 2 + by 2 = z. Cutting by a plane z = d yields an ellipse when d > 0 and the empty set when d < 0. Cutting by the plane x = 0 yields the parabola by 2 = z in the (y, z)-plane and analogously when cutting by the plane y = 0. The surface has the shape depicted in Fig. 34 and is called an elliptic paraboloid. • ax 2 − by 2 = z. Cutting by a plane z = d always yields a hyperbola; the foci are in the direction of the x-axis when d > 0 and in the direction of the y-axis when d < 0. Cutting by the plane z = 0 yields √ √ √ √ ( ax + by)( ax − by) = 0 42 1 The Birth of Analytic Geometry Fig.