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p p If u sin 1 x sin 1 y and v x 1 y2 y 1 x2 , determine whether there is a functional relationship between u and v, and if so nd it. If F xy yz zx, G x2 y2 z2 , and H x y z, determine whether there is a functional relationship connecting F, G, and H, and if so nd it. Ans. H 2 G 2F 0. @ x; y; z @ u; v; w 1 provided (a) If x f u; v; w , y g u; v; w , and z h u; v; w , prove that @ u; v; w @ x; y; w @ x; y; z 6 0. (b) Give an interpretation of the result of (a) in terms of transformations. @ u; v; w If f x; y; z 0 and g x; y; z 0, show that dx dy dz @ f ; g @ f ; g @ f ; g @ y; z @ z; x @ x; y giving conditions under which the result is valid. @x @2 x ; b ; @u @u2 de ne x; y; and z as twice di erentiable functions of u, v; and w. (a) a 1 ; 1 8xyz b 16x2 y 8yz 32x2 z2 ; 1 8xyz 3 c @2 x assuming that the equations @u @v

The relative maximum points are (0.354431, 1.33423), (1.61107, 2.59086), and (2.8677, 3.8475). The relative minimum points are (0.902206, 0.0775897) and (2.15884, 1.17905). Note: Caution must be taken to examine the results of the calculation. The value obtained is not necessarily the one closest to the initial guess. For example, FindMaximum[f[x], {x, 2.8}]

4.12434}}

8.16 Find two positive numbers whose sum is 50, such that the square root of the first added to the cube root of the second is as large as possible.

Ans: 6.95.

y = 50 x; f[x_]= x + 3 y; Plot[f[x],{x, 50}];

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# Put a capping on the size of the log files (in Kb). max log size = 50 # Security mode. Most people will want user level security. See # security_level.txt for details. security = user # Use password server option only with security = server ; password server = <NT-Server-Name> # You may wish to use password encryption. Please read # ENCRYPTION.txt, Win95.txt and WinNT.txt in the Samba documentation. # Do not enable this option unless you have read those documents ; encrypt passwords = yes # Using the following line enables you to customise your configuration # on a per machine basis. The %m gets replaced with the netbios name # of the machine that is connecting ; include = /usr/local/samba/lib/smb.conf.%m # Most people will find that this option gives better performance. # See speed.txt and the manual pages for details socket options = TCP_NODELAY # Configure Samba to use multiple interfaces # If you have multiple network interfaces then you must list them # here. See the man page for details. ; interfaces = 192.168.12.2/24 192.168.13.2/24 # Browser Control Options: # set local master to no if you don't want Samba to become a master # browser on your network. Otherwise the normal election rules apply ; local master = no # OS Level determines the precedence of this server in master browser # elections. The default value should be reasonable ; os level = 33 # Domain Master specifies Samba to be the Domain Master Browser. This # allows Samba to collate browse lists between subnets. Don't use this # if you already have a Windows NT domain controller doing this

8 7 6 5

NSolve[f'[x] 0]

State and prove a theorem similar to that in Problem 6.35, for the case where u f x; y; z , v g x; y; z , w h x; y; z .

41.1553}}

- 218 -

y /. x 41.1553 8.8447 f[41.1553] 8.48329 The two numbers are x = 41.1553 and y = 8.8447. The maximum sum is 8.48329.

8.17 A right circular cylinder is inscribed in a unit sphere. (a) Find the largest possible volume. (b) Find the largest possible surface area.

TRANSFORMATIONS, CURVILINEAR COORDINATES 6.96. Given the transformation x 2u v, y u 3v. (a) Sketch the region r 0 of the uv plane into which the region r of the xy plane bounded by x 0, x 1, y 0, y 1 is mapped under the transformation. @ x; y (b) Compute . (c) Compare the result of (b) with the ratios of the areas of r and r 0 . @ u; v Ans. (b) 7

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