# Advanced calculus. Problems and applications to science and by Hugo. Rossi

By Hugo. Rossi

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Choose a > 0 so that the 2a-neighborhood of (0, 0) is contained in E and hence so is the the triangle with vertices O, Q : (a, 0), R : (0, a). For each point (x, y, / ( x , y)) for which (x, y) is on the segment QR, join the point to the origin by a line segment (a chord of the graph of / ) . By the continuity of convex functions of one variable, / is continuous along QR, and hence the surface formed by the chords drawn is the graph of a continuous function #(x, y). By convexity/(x, y) < g(x, y) in the triangular region.

16). 23 Problems in Two Dimensions Saddle Points If at a critical point of / one has ac — b2 < 0, then the corresponding quadratic function Q takes on both positive and negative values on the unit circle. One verifies that / can have neither a maximum nor a minimum at the critical point. One says that / has a "saddle point" at the critical point. This case is illustrated by the function f(x, y) = x2 — y2. ) Positive Definite Quadratic Functions A quadratic function Q(u, v) = au2 + Ibuv + cv2 is said to be positive definite if Q > 0 for all (u, v) except (0, 0).

B¿. Here k is the nullity of the matrix A; k — n — r, where r is the rank of A, the maximum number of linearly independent columns (or rows) of A. We give a simple example. Example 1. The equations are Χγ -f χ2 = 1, X2 + X3 = 2, *3 + X4 = 3. One can write the solutions as x\ = £, x2 = 1 - t, x¿ = 1 + í, X4 = 2 - t or as the vector equation x = (0, 1, 1, 2)f + i(l, - 1 , 1, - 1 ) ' . 19. Geometry of a plane in space. Geometry of //-Dimensional Space 35 The matrix A has rank 3 and nullity 1.