1) Show that the force response of a MDOF system with general damping can be written as: X liax) -Σ = ral iw-s, + {0} iw-s,

If A = yz(^2) i - 3xz(^2) j + 2xyz k , B = 3x i + 4z j - xy k and ∅= xyz , find (a) Ax (∇∅), (b) (Ax ∇) ∅, (c) (∇x A) x B, (d) B. ∇x A.

Match each of the functions (a)-(c) with the corresponding gradient vector field from those in (I)-(VI). Enter your answer as upper case roman numerals: I, II, III, etc. (a) f(x, y) = x cos(y) (b) f(x, y) = (x - 2)²(y-2)/5 (c) f(x, y) = sin(x) · cos(y) (click on an image to enlarge it) IV II V II VI

Let q = - 2xy - y² + 2xz+2yz + z² be a quadratic form on R³ viewed as a polynomial in 3 variables. Find a linear change of variables tou, v, w that puts q into the canonical form in `Sylvester's law of inertia'!. What are the values of the associated indices s, t? Select one: O We let u = √3(x − y + 2), v = x+y, w = (x - y)/2 to find that q = u²+². Hence s = 2, t = 0 in Sylvester's law of intertia. O We let u = x - 2y, v = z+y, w = √3y to find that q=u²v² w². Hence s = 1,t = 2 in Sylvester's law of intertia. O we let u = x, v= √2(x+y), w = x+y+z to find that q = u²v² + w²2. Hence s = 2, t = 1 in Sylvester's law of intertia. O None of the others apply O The quadratic form does not obey the condition to be diagonalisable over R. This is because the minimal polynomial of the corresponding matrix is not a product of distinct linear factors. Hence Sylvester's law of inertia does not apply. By convention, we set s = t = ∞ when this happens.

3. (12) Let V = R². Define vector addition and scalar multiplication as follows: (u1, uz)O(v1, v2) = (u1V1, U2V2) and c(u1,u2) = (u,C, u2^), c > 0 (a) Does V have an additive identity? If so, what would it be? (b) Does the distributive axiom c(uOv) = cu@cv hold? (Show why or why not). 4

Section 17.7) Verify Stokes' Theorem for F(x, y, z) = if C' is the intersection of z = 4 + y and x² + y² = 4 counterclockwise when viewed from above.

1. (Section 17.1) Match the vector fields with their sketches below by placing the letter of the function in the corresponding blank: (I) F(x, y) = (II) F(x, y) = (III) F(x, y) = (IV) F(x, y) = Vector Field: (B) Vector Field: (A) (C) Vector Field (D) 0 Vector Field:

Draw approx. solution for the directional field for the first order with the initial condition y(0) = -1.8

Show whether the following functions are wave functions or not. 1. У(х, t) еxp(ikx) = A- exp (i(ot-Ф)) кЗх3-0313-3kоxt(kx-ot)-iф)) 2 У(х, t) 3. y(x, t) Aexp(i(k³x³-w³t³-3kwxt(kx-wt)-ip)) Аехр (i(-kx? + оt))

Let a=<−5,0,−3> and b=<4,0,2> Show that there are scalars s and t so that sa+tb=<27,0,15>.s = ? t = ?