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Chapter 13: Q 71. (page 1066)

Prove Theorem $13.2 (a)$ . That is, prove that ${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} a_{j k} = {鈭�}_{k = 1}^{n} {鈭�}_{j = 1}^{m} a_{j k}$

Short Answer

Expert verified

This is proved using expansion of left hand side of the equation.

Step by step solution

01

Given Information

We need to prove that

${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} a_{j k} = {鈭�}_{k = 1}^{n} {鈭�}_{j = 1}^{m} a_{j k}$

02

Simplification

Expansion gives ${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} a_{j k} = {鈭�}_{j = 1}^{m} ({鈭�}_{k = 1}^{n} a_{j k})$

$= {鈭�}_{j = 1}^{m} (a_{j 1} + a_{j 2} + a_{j 3} + 鈥� + a_{j n})$

$= (a_{11} + a_{12} + a_{13} + 鈥� + a_{1 n}) + (a_{21} + a_{22} + a_{23} + 鈥� + a_{2 n}) +$

$+ (a_{31} + a_{32} + a_{33} + 鈥� + a_{3 n}) + 鈥� + (a_{m 1} + a_{m 2} + a_{m 3} + 鈥� + a_{m n})$

Grouping gives

${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} a_{j k} = (a_{11} + a_{12} + a_{13} + 鈥� + a_{1 n}) + (a_{21} + a_{22} + a_{23} + 鈥� + a_{2 n}) +$

$+ (a_{31} + a_{32} + a_{33} + 鈥� + a_{3 n}) + 鈥� + (a_{m 1} + a_{m 2} + a_{m 3} + 鈥� + a_{m n})$

$= (a_{11} + a_{21} + a_{31} + 鈥� + a_{m 1}) + (a_{12} + a_{22} + a_{32} + 鈥� + a_{m 2}) +$

$+ (a_{13} + a_{23} + a_{33} + 鈥� + a_{m 3}) + 鈥� + (a_{1 n} + a_{2 n} + a_{3 n} + 鈥� + a_{m n})$

${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} a_{j k} = (a_{11} + a_{21} + a_{31} + 鈥� + a_{m 1}) + (a_{12} + a_{22} + a_{32} + 鈥� + a_{m 2}) +$

$+ (a_{13} + a_{23} + a_{33} + 鈥� + a_{m 3}) + 鈥� + (a_{1 n} + a_{2 n} + a_{3 n} + 鈥� + a_{m n})$

$= {鈭�}_{k = 1}^{n} (a_{1 k} + a_{2 k} + a_{3 k} + 鈥� + a_{m k})$

$= {鈭�}_{k = 1}^{n} {鈭�}_{j = 1}^{m} a_{j k}$

Hence ${鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} a_{j k} = {鈭�}_{k = 1}^{n} {鈭�}_{j = 1}^{m} a_{j k}$

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Most popular questions from this chapter

$How many summands are in {鈭�}_{j = j_{0}}^{m} {鈭�}_{k = k_{0}}^{n} j^{k} ?$

Evaluate each of the double integral in the exercise 37-54 as iterated integrals

$鈭� {鈭�}_{R} (y^{2}) d A W h e r e R = {(x, y) | - 3 鈮� x 鈮� 2, - 2 鈮� y 鈮� 2}$

How many summands are in ${鈭�}_{i = i_{0}}^{l} {鈭�}_{j = j_{0}}^{m} {鈭�}_{k = k_{0}}^{n} k^{i} j^{k}$ ?

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{0}^{3} {鈭�}_{0}^{3} {鈭�}_{0}^{3 - y} f (x, y, z) d z d y d x$

In Exercises 57鈥�60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.

Assume that the density ofR is uniform throughout.

(a) Without using calculus, explain why the center of mass is (2, 3/2, 1).

(b) Verify that the center of mass is (2, 3/2, 1), using the appropriate integral expressions.

See all solutions

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