Tutorial ODWE: One-Dimensional W Equation ave Thus, @2 f 1 @2 f d2f d2f 2 2 = 2 2 = 0: @x2 c @t du du The proof is similar for f (u+) : Exercise 10. See, for example, Fishbane, Gasiorowicz and Thornton, `Physics for Scientists and Engineers (Extend
Tutorial V AM: V ector Algebra and an Introduction to Matrices = = = = = (ijk jnp) lmi al bm cndp (ipkn in kp) lmial b mc ndp lmial bm (ck di ci dk ) (A B)i (c k di c idk ) [ (A B) D]ck [(A B) C]dk:
Exercise 19. If A; B and C are not linear
Tutorial FODE: Ordinary First-Order Differential Equations p b. The equation is scale invariant for n = 1=2 = y = v= x =) ) 2 v + 1 dv 3 dx = : 2 1) v (v 2 x A second change of variable, z = v 2; gives 2 1 dx dz = 3 ; z1 z x which integrates to
Tutorial FODE: Ordinary First-Order Differential Equations yields 1 y (x) = Ce3x + x : 3 The condition gives 3 = C 1=3 = C = 10=3; or ) y (x) = 1 3x 10e + 3x 1 : 3
b. Take yp = Aex: Then, Aex 2Aex = ex =) A = 1 or A = 1: The homogenous solutio
Tutorial V AM: V ector Algebra and an Introduction to Matrices summation convention, we simply drop the sum sign, and write ai ij = aj : Exercise 14. A B = aibi = ai (b j ij ) = ai bj ij : Exercise 15. Recall that the summation convention is in effe
Tutorial V AM: V ector Algebra and an Introduction to Matrices
9.
Solutions to Exercises and Problems
Exercise 1. jA Bj = AB jcos j AB since jcos j 1: p p p Exercise 2. jA + Bj = (A + B) (A + B) = A2 + B 2 + 2A B A2 + B 2 + 2AB = q (A + B)2
Tutorial CA: A Workbook on Complex Arithmetic
9.
Solutions to Exercises and Problems
a. x2 = 9 =) x = 3i p b. x2 = 17 =) x = 17i a. b. c. d. = i2 i = i = i4 i2 = 1 (1) = 1, since i4 = i2 i2 = (1) (1) = 1: = i4 i3 = 1 (i) = i 2 = i4 i = 12
Tutorial SOLDE: Second-Order Linear Differential Equations The solution is thus p p 2 t 9 3 y (t) = e cos 5t + p sin 5t + (t sin 2t 2t cos 2t sin 2t + 2cos 2t) : 5 10 5 Exercise 23 NA Exercise 24 From Newton's Second Law, the equation of motion
Tutorial V AM: V ector Algebra and an Introduction to Matrices Exercise 62. a. In a rotation about the x-axis, only the y and z components are changed. Thus, the y-axis is turned toward the z-axis, and so forth. So, 0 1 1 0 0 R1 ('1) = @ 0 cos '1 si
Tutorial V AM: V ector Algebra and an Introduction to Matrices n2 r =17x + 19y + 20z = n2 A =75: The equations describing the line can thus be expresses parametrically as x = t; y = 5 3t; z = 1 + 2t: p p The distance from the origin to the line is
Tutorial V AM: V ector Algebra and an Introduction to Matrices Row 3 by 2=39: 0 1 0 1 4 0 0 1 0 0 @ 0 1 0 A ! @ 0 1 0 A =) 2 0 0 39 0 0 1 2
0 @
20 13 2 13 3 2
which = B1;as can be checked by multiplying. Exercise 43. We have jBj = 39: Computing t
Tutorial FODE: Ordinary First-Order Differential Equations
6.
Solutions to Exercises
Exercise 1. The antiderivative of 2x is x2: Thus, the general solution is y (x) = x2 + C:
The condition tells us 0 = 12 + C =) C = 1; or y (x) = x2 1: Exercise 2
Tutorial SOLDE: Second-Order Linear Differential Equations The general solution is thus y(x) = cos x + sin x + + cos x [(10x + 3) sin x + (5x + 4) cos x] e2x 25
sin x [(5x + 4) sin x (10x + 3) cos x] e2x 25 5x + 4 2x = cos x + sin x + e : 25 c.
Tutorial FODE: Ordinary First-Order Differential Equations y (x) =
1 2
y (0) = 1 = C = 3=2 =) )
2 x2 1 ex + C 1 2 2 = x 1 + ex C: x2 e 2 1 2 2 x 1 + 3ex : 2
y (x) =
(In the integral for v (x), make the change of variable z = x2; and integ
Tutorial MT: Matrix Theory
5.
Solutions to Exercises
Exercise 1.
1. From the definition of the scalar product, Eqn. 16, using the summation convention, (A; B) = (abi ) = aib = b ai = (B; A). i i i P 2. (A; A) = a ai = n jaij2 0, since it is the
Tutorial MT: Matrix Theory = I + R + R2 + R3 + + Rn + S
= S + RS + R2S + :
The value of this solution will depend upon its convergence, in a sense that relates to the matrices involved. Exercise 8. One can see by inspection that the Paul
Tutorial MT: Matrix Theory 2 4 3 = I 1 + + ii + 2! 4! 3! = I cos + ii sin : Exercise 16. We have ^ = (sin cos ') 1 + (sin sin ') 2 + (cos ) 3 ; so that r h i (^ )2 = (sin cos ')2 + (sin sin ')2 + cos2 I r + sin2 cos ' sin ' [1
Tutorial ODWE: One-Dimensional W Equation ave
5.
Solutions to Exercises
Exercise 1. Transverse waves have the disturbance perpendicular to the direction of propagation of the disturbance. Don't confuse the wave motion with the propagation. Exercis
Tutorial OFFS: Orthogonal Functions and Fourier Series
3.
Solutions to Exercises
Exercise 1. Refer to Eqns. 19 - 21 in the tutorial, TF: ` Trigg' Functions. Cn = =2: ' Exercise 2. One might say, If there exists no vector V, except the null vector,
Gram-Schmidt Process for Hermite and
Laguerre Polynomials
Brian Cosey
Mihai Gheorghe
Brian Willhelm
Weston Wands
Spencer Davidson
Ellen Bulka
For the first problem, we wish to construct the first five orthogonal polynomials from the set
cfw_xn, n = 0, 1,