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GEOMETRIC PROPERTIES 1. Two vectors A and B are equal if they have the same magnitude and direction regardless of their initial points. Thus A B in Fig. 7-1 above. In other words, a vector is geometrically represented by any one of a class of commonly directed line segments of equal magnitude. Since any one of the class of line segments may be chosen to represent it, the vector is said to be free. In certain circumstances (tangent vectors, forces bound to a point), the initial point is xed, then the vector is bound. Unless speci cally stated, the vectors in this discussion are free vectors. A vector having direction opposite to that of vector A but with the same magnitude is denoted by A [see Fig. 7-2]. 150

9 10 7 1

Since the determinant is 0, we anticipate either no solution or an infinite number of solutions. Since the null space contains a non-zero vector, there will be an infinite number of solutions.

- 221 -

generalsolution = t * Flatten[nullspacebasis] + particular 4 { 20 13 t, 7 + 11 t, 17 10 t,7 t} 7 7

SURFACE INTEGRALS 10.52. (a) Evaluate x2 y2 dS, where S is the surface of the cone z2 3 x2 y2 bounded by z 0 and z 3.

Now substitute the (positive) value of h into the formula for s: s[r_]= 2 o r h + 2 o r2 /. h 2 1 r2 2 r2 + 4 r 1 r2 Solve for the critical value of r : Solve[s'[r] 0,r] 1 1 r 10 5 + 5 , r 10 5 5

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Only the positive value of r is acceptable. We use it to compute the maximum surface area. s 1 5 + 5 //Simplify 10

(1 + 5 )

(b) Interpret physically the result in (a).

# A publicly accessible directory, read/write to all users. Note that all files # created in the directory by users will be owned by the default user, so # any user with access can delete any other user's files. Obviously this # directory must be writable by the default user. Another user could of course # be specified, in which case all files would be owned by that user instead. ;[public] ; path = /usr/somewhere/else/public ; public = yes ; only guest = yes ; writable = yes ; printable = no # The following two entries demonstrate how to share a directory so that two # users can place files there that will be owned by the specific users. In this # setup, the directory should be writable by both users and should have the # sticky bit set on it to prevent abuse. Obviously this could be extended to # as many users as required. ;[myshare] ; comment = Mary's and Fred's stuff ; path = /usr/somewhere/shared ; valid users = mary fred ; public = no ; writable = yes ; printable = no ; create mask = 0765

Sign s' 1 5 + 5 ' 10 1

8.18 Find the points on the circle x 2 + y 2 2 x 4 y = 0 closest to and furthest from P(4, 4).

Ans:

First we draw a diagram. circle = ContourPlot[x2 + y2 2 x 4 y 0, {x, 5, 5}, {y, 2, 5}]; point = Graphics[{PointSize[Medium], Point[{4,4}]}]; Show[circle, point, Frame False, AspectRatio Automatic, Axes True]

5 4 3 2 1 4 2 1 2 2 4

- 19 -

Let (x, y) represent a point on the circle. First, we need to solve for y in terms of x. Solve[x 2 + y 2 2 x 4 y 0, y]//Simplify

10.53. Determine the surface area of the plane 2x y 2z 16 cut o by (b) x 0; y 0, and x2 y2 64. Ans: a 9; b 24

We shall minimize the square of the distance from (x, y) to (4, 4). We call this d2. It is clear from the picture that the point closest to P lies on the upper semicircle. y = 2 + 4 + 2 x x2; d2[x_]=(x 4)2 +(y 4)2; Solve[d2'[x] 0] 1 13+ 3 65 x 13

{2.86052,

3.24035}

# 127.0.0.1 128.95.135.13 128.95.135.24 128.95.142.30 140.27.133.4

10.54. Find the surface area of the paraboloid 2z x2 y2 which is outside the cone z p Ans: 2 5 5 1 3

The point furthest from P lies on the lower semicircle. y = 2 4 + 2 x x2; d2[x_]=(x 4)2 +(y 4)2; Solve[d2'[x] 0]

x 1 (13 3 65 ) 13

5 , 2 2 5 13 13

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