1. Substituting E(r,t) = (1, 0, 0)E, exp[i(kyy+kız – wt)] into Helmholtz equation: V’E(r,t) = (us)ə’E(r,t)/ət?,...
(i) Consider the wave Ē(7,t) = Ło cos(wt – k ), where Ē, is a fixed vector. Determine the relation between w and k = \KI SO that Ē(7,t) is a solution of the wave equation -27 182 VPE = 2 ət? - What is the direction of propagation of the wave? ii) Show, by substitution of Ē(7,t) in the appropriate Maxwell's equation, that K· Ē= 0. iii) Assuming that the magnetic field B(7, t) = B, cos(wt – K:1),...
3. Let W (x, t) = (coswt)(a cos nx+b sin nx) and (x, t) = (exp -kn-t)(a cos nx+ bsin nx). Here n is a positive integer, 2,t are real variables, and a, b,w, k are real constants with k positive. a. Evaluate W(x,0), H (2,0) and əW/ət(x,0) for all c. b. Show OH/ət = k(32H/8x2) for all x,t. c. Find some positive constant c so that w2w/at2 = c(32W/8x?) for all x, t.
(Exponential martingales) Suppose O(t,w) = (01(t, w),...,On(t,w)) E R" with Ox(t,w) E VIO, T] for k = 1,..., n, where T < 0o. Define 2. = exp{ jQ1, wydBlo) – 4 640,w.do}osist where B(s) ER" and 62 = 0 . 0 (dot product). a) Use Ito's formula to prove that d24 = 2:0(t,w)dB(t). b) Deduce that 24 is a martingale for t <T, provided that Z40x(t,w) € V[O,T] for 1 sk sn.
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C (b) Prove that when z є R, the definition of exp z given above is consistent with the one given in problem (2a), assignment 16. Definition from Problem (2a): L(x(1/t)dt E(z) = L-1 (z)
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C...
Needs to prove the second picture.
n 1 exp(1) = lim นม โO E! k=0 Prove there exists a continuous function exp : R → R.
R t = i(t) C 2011P E e p gP a P9.09 10ed The voltage applied to this circuit at t 0 (when the switch closes) is v (t) = 75 cos (4,000t - 60°) Volts Also given that R = 400 2 (0hm) and L=75 mH (milli Henry) The initial inductor current is zero for t< 0 The textbook gives you the total response equation as: )_ ?(0-¢)so R2+(w L) Cos(wt+¢-e) -V V m i(t)=itransient(t)+isteady.state(t)=R2 +(wL m - ㅎCOS...
Given the position vector r(t), determine v,lv. a, T,K : r = r(t) (1 + et)i + e
Given the position vector r(t), determine v,lv. a, T,K : r = r(t) (1 + et)i + e
1 1 0 -1 Exercise 2. Let A = 0 1 0 in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R”, set g(ū) = WT AV E R. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]c,b in the bases C = {1} and B = { 9 8 B |}? (ii) Let f: R3 + R be the function defined by f(w) = vſ Aw...
2. The electric field in a plane wave is described by the equation (k > 0): Ē(x,y,z,1)= E, sin(kz – mt)ị Answer the following questions about the wave. i. What direction is the wave traveling? Explain how you can tell from the equation for the electric field. ii. Write an expression for the magnitude of the magnetic field of the wave. iii. Calculate the average intensity of the wave if Eo = 3000 V/m. The MKS units of intensity are...
o 1 0 -1 Exercise 2. Let A= in M3,R, and ✓ = 0 in R3. -1 0 For every vector W E R3, set g(W) = WT AT ER. (i) Show that g: R3 → R defines a linear transformation. What is the matrix [g]C,B in the - 1 bases C = {1} and B { 8.00 } ? (ii) Let f : R3 → R be the function defined by f() = 7T Aw E R. Show that...