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Let u = (a, b, c) and v = (d, e, f) and compute w = u v. Then verify that w^u and w^v. Two vectors are orthogonal (^) if their dot product is 0. u = {u1, u2, u3}; v = {v1, v2, v3}; w = Cross[u, v]

u3 v2 + u2 v3, u3 v1 u1 v3, u2 v1 + u1 v2}

Ans:

12.10 It can be shown that the volume of a parallelepiped formed by u, v, and w is | u (v w) |. Compute the volume of the parallelepiped formed by i + 2 j 3 k, 2 i 5 j + k , and 3 i + j + 2 k. (The quantity u (v w) is called the scalar triple product.)

u = {1, 2, 3}; v = {2, 5, 1}; w = {3, 1, 2}; volume = Abs[u.Cross[v, w]] 64

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14.26. (a) Show that the solution to Problem 14.13 can be written p p 2 x=2 t v2 1 1 x =2 t v2 e dv p e dv U x; t p 0 1 x =2p t

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12.11 Let u = (u1, u2, u3), v = (v1, v2, v3), and w = (w1, w2, w3). Prove that the scalar triple product, u1 u2 u3 u (v w) = v1 v2 v3 . w1 w2 w3

u = {u1, u2, u3}; v = {v1, v2, v3}; w = {w1, w2, w3}; matrix = {{u1, u2, u3}, {v1, v2, v3}, {w1, w2, w3}}; lhs = u.Cross[v, w] // Expand; rhs = Det[matrix] // Expand; lhs rhs True

hilbert = HilbertMatrix[6]; hilbert // MatrixForm 1 1 2 1 3 1 4 1 5 1 6 1 2 1 3 1 4 1 5 1 6 1 7 1 3 1 4 1 5 1 6 1 7 1 8 1 4 1 5 1 6 1 7 1 8 1 9 1 5 1 6 1 7 1 8 1 9 1 10 1 6 1 7 1 8 1 9 1 10 1 11

@U @2 U 2 and the conditions of Problem 14.13. @t @x & 1 jxj < 1 14.27. Verify the convolution theorem for the functions f x g x . 0 jxj > 1 (b) Prove directly that the function in (a) satis es 14.28. Establish equation (4), Page 364, from equation (3), Page 364. 14.29. Prove the result (12), Page 365. 1 1 f u ei u du and Hint: If F p 2 1 F G ! Now make the transformation u v x: 1 F G p De ne 1 f g p then 1 F G p 1 f u g x u du

A long time ago, in a LAN far, far away, an API was developed to extend the Basic Input/Output System (BIOS) services of personal computers to include support for sharing information between computers over a network. The resulting NetBIOS API is documented in the IBM PC Network Technical Reference Manual, (September 1994). The NetBIOS API defines three service types: session service, datagram service, and name service. Conversations between NetBIOS applications are facilitated by NetBIOS session and datagram services. The NetBIOS session service is used when a reliable virtual circuit with guaranteed delivery is required for passing messages between two known parties. An example of a session-based connection would be accessing a file share. In contrast, the NetBIOS datagram service is used for connectionless communication without guaranteed delivery when the target recipient is not known in advance. Datagrams, or Mailslots in the Microsoft parlance, are usually transmitted as network broadcasts or multicasts to a group of machines. Datagram services are used when a client requests the identification of the master browser.

Det[hilbert] 1 186 313 420 339 200 000

MatrixForm[Inverse[hilbert], TableAlignments Right]

36 630 3 360 7 560 7 560 -2 772 630 14 700 -88 200 211 680 220 500 83 160 564 480 1 411 200 1 512 000 0 582 120 3 360 88 200 7 560 211 680 1 411 200 1 3 628 800 3 969 000 1 552 320 1 512 000 3 969 000 4 410 000 1 746 360 7 560 220 500 2 772 83 160 582 120 1 552 320 1 746 360 0 698 544

1 g v ei v dv,

TableForm[Table[{k, Det[HilbertMatrix[k]] // N}, {k, 1, 10}], TableSpacing {1, 5}, TableHeadings {None, {"k", "determinant"}}]

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