Mathematical Methods in the Physical Sciences

3rd Edition

ISBN: 9780471198260

Author: Mary L. Boas

Publisher: Wiley, John & Sons, Incorporated

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Textbook Question

Chapter 6.7, Problem 8P

Compute the divergence and the curl of each of the following vector fields.

V = sinh z i + 2 y j + x cosh z k

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Chapter 6.7, Problem 7P

Chapter 6.7, Problem 9P

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3. Let Pk denote the vector space of all polynomials with degree less than or equal to k. Define a linear transformation T : P₁ → P3 by T(f(x)) = ƒ(0)+ƒ'(1)(x − 1) + ƒ"(2)(x − 2)² + ƒ" (3)(x − 3)³. Find the matrix representation for T relative to the standard basis {1, x, x2, x3, x} of P4 and the reversed standard basis {x³, x², x, 1} of P3.

1. Let P denote the vector space of all polynomials and f'(x) denote the derivative of f = P. State "True" or "False" for each of the following statements. You do not need to justify your answers. (1) T R2 R3 defined by T(x, y) = (x²x, 0, y/2) is a linear transformation. (2) T: R² → R³ defined by T(x, y) = (x+y, x―y, xy) is a linear transformation. (3) T R2 R3 defined by T(x, y) = (1,0,0) is a linear trans- formation. (4) T R2 R3 defined by T(x, y) = (y, x, y) is a linear trans- formation. (5) TR² R³ defined by T(x, y) transformation. (6) T: R2R3 defined by T(x, y) is a linear transformation. = = (x, x2, x3) is a linear (2x+3y, 3x+4y, 4x+5y) (7) T: PP defined by T(f(x)) = ƒ(0) + ƒ′(0)x + ƒ"(0)x² is a linear transformation. (8) T: P→ P defined by T(f(x)) = ƒ(1) + ƒ'(2)x + ƒ"(3)x² is a linear transformation. (9) TPP defined by T(f(x)) = f(x-3) is a linear trans- formation. (10) TPP defined by T(f(x)) = f(x) - 3 is a linear trans- formation.

2. Find a basis for the kernel of the linear transformation T: R5 → R³ defined by T(x) = Ax, where vectors in R³ and R5 are written as columns, 1 2 -3 4-5 A = 1 -1 0 1 -2 1 0 1 0 -1 and Ax denotes matrix multiplication. Show all your work! continued on next page!!!! 1

Chapter 6 Solutions

Mathematical Methods in the Physical Sciences

Ch. 6.3 - If A=2ijk,B=2i3j+k,C=j+k, find (AB)C,A(BC),(AB)C,...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - For Problems 2 to 6, given A=i+j2k,B=2ij+3k,C=j5k:...Ch. 6.3 - A force F=2i3j+k acts at the point (1,5,2). Find...Ch. 6.3 - Prob. 8P Ch. 6.3 - Prob. 9P Ch. 6.3 - In Figure 3.5, let r be another vector from O to...

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