(b) Beweisen Sie, dass eine Abbildung d: A× A →→→→R genau dann eine Metrik auf der Menge A ist, wenn (4 P.) Vx, yЄA: d(x,y) = 0 ⇒ x=y (1) und Vx, y, z ЄA: d(x, y) ≤ â(z, x) + d(y, z) (2) gelten.

Two surfaces S and S^(-) with a common point p have contact order ≥ 2 at p if there exist parametrization x(u,v) and x^(-)(u,v) in p of S and S^(-) respectively such that xu = x^(-)u, xv = x^(-)v, xuu = x^(-)uu, xuv = x^(-)uv, xvv = x^(-)vv at p. Prove the following: a. Let S and S^(-) have contact order greater than or equal to 2 at p; x:U -> S and x^(-): U -> S^(-) be arbitrary parametrizations in p of S and S^(-) respectively and f: V c R^(3) -> R be a differentiable function in a neighborhood V of p in R^(3). Then the partial derivatives of order smaller than or equal to 2 of f o x^(-): U -> R are zero in x bar^(-1)(p) iff the partial derivatives of order smaller than or equal to 2 of f o x: U -> R are zero in x^(-1) (p). b. Let S and S^(-) have contact of order smaller than or equal to 2 at p. Let z = f(x, y), z = f^(-) (x, y) be the equations in a neighborhood of p, of S and S^(-) respectively where the xy plane is the common tangent plane at p = (0, 0). Then the…

Consider the Cauchy Problem y 0 = a(x) arctan y, y(0) = 1, where a(x) is a continuous function defined on R, such that for every x it holds that |a(x)| ≤ 1. Using the Global Picard–Lindel¨of Theorem, show that there exists a unique solution y defined on R.

If x, y, and z are lengths of the edges of a rectangularbox, then the common length of the box’s diagonals iss = √(x2 + y2 + z2). Assuming that x, y, and z are differentiable functions of t,how is ds/dt related to dx/dt, dy/dt, and dz/dt?

Verify Stokes' Theorum for F(x,y,z) = (y,-x,yz) if S is the paraboloid z = x2 + y2 with x2 + y2 = 1, z=1 as its boundary.

Prove that G = <x, y | x2 = yn= e, xyx = y-1> is isomorphic to Dn.

Suppose masses m1, m2, m3, m4 are located at positions x1, x2, x3, x4 in aline and connected by springs with spring constants k12, k23, k34 whose naturallengths of extension are l 12, l 23 , l 34• Let / 1, / 2, / 3 , / 4 denote the rightwardforces on the masses, e.g., / 1 = k12(x2 - x1 -i12). 10 PART I. FUNDAMENTALS(a) Write the 4 x 4 matrix equation relating the column vectors I and x. LetK denote the matrix in this equation.(b) What are the dimensions of the entries of K in the physics sense (e.g.,mass times time, distance divided by mass, etc.)?(c) What are the dimensions of det(K), again in the physics sense?(d) Suppose K is given numerical values based on the units meters, kilograms,and seconds. Now the system is rewritten with a matrix K' based on centimeters, grams, and seconds. What is the relationship of K' to K? What is therelationship of det(K') to det(K )?