FIGURE 1.8 Regular polygon of n sides. I.r=b cscI FIGURE 1.9 Right circular cylinder. V=TRh lateral surface area= 2IT Rh
12 h R Figure 1.9 Right circular cylinder. V R h = π 2 ; lateral surface area = 2π Rh. R b θ Figure 1.8 Regular polygon of n sides. A n b n = 4 2 ctn ; π R n = b 2 csc . π
FIGURE 1.10 Cylinder (or prism) with paralle bases. V=Ah FIGURE lll Right circular cone. V=-tRh lateral surface area =TRI=TRVR+h23
13 R l h Figure 1.11 Right circular cone. V R h = 1 3 2 π ; lateral surface area = = + π π Rl R R h 2 2 . h A Figure 1.10 Cylinder (or prism) with parallel bases. V = Ah
FIGURE 1. 12 Sphere. V=-TR: surface area 4πR2
14 R Figure 1.12 Sphere. V R = 4 3 3 π ; surface area = 4π R2
Determinants. matrices and linear systems of equations I. Determinants Definition. The square array(matrix)A, with n rows and n columns. has associated with it the determinant a21 a2 a number ∑(±)a1a2ax…au where i, j, k,..., I is a permutation of the n integer 1, 2,. ., n in some order. The sign is plus if the permutation is even and is minus if the permutation is odd(see 1. 12). The 2 x 2 determinant (1, 2) is even and (2, ) is odd once the permutation permutations are as follows
15 2 Determinants, Matrices, and Linear Systems of Equations 1. Determinants Definition. The square array (matrix) A, with n rows and n columns, has associated with it the determinant det A a a a a a a a a a n n n n nn = … … … … … … … 11 12 1 21 22 2 1 2 , a number equal to ∑ ( ) ± … a a a a 1 2 3 i j k nl where i, j, k, …, l is a permutation of the n integers 1, 2, 3, …, n in some order. The sign is plus if the permutation is even and is minus if the permutation is odd (see 1.12). The 2 × 2 determinant a a a a 11 12 21 22 has the value a a a a 11 22 12 21 − since the permutation (1, 2) is even and (2, 1) is odd. For 3 × 3 determinants, permutations are as follows:
1. 2. 3 even 1,3,2odd 3. 1. 2 even 3,2,1odd Thus a determinant of order be the sum of n! signed produc 2. Evaluation by Cofactors Eachelement a; has a determinant of order(n-1 minor(Mi obtained by suppressing all eler row i and column j. For example, the minor of a,, in the 3 x 3 determinant above is
16 1, 2, 3 even 1, 3, 2 odd 2, 1, 3 odd 2, 3, 1 even 3, 1, 2 even 3, 2, 1 odd Thus, a a a a a a a a a a a a a 11 12 13 21 22 23 31 32 33 11 22 33 1 = + − . . 1 23 32 12 21 33 12 23 31 13 21 . . . . . . . . a a a a a a a a a a − + + a a a a 32 13 22 31 − . . A determinant of order n is seen to be the sum of n! signed products. 2. Evaluation by Cofactors Each element aij has a determinant of order (n – 1) called a minor (Mij) obtained by suppressing all elements in row i and column j. For example, the minor of element a22 in the 3 × 3 determinant above is a a a a 11 13 31 33