1. The air pressure in a particular room varies from point to point and is given by p(x, y, 2) = 3x + xỉ exp(-2 y), where (x, y, z) describes a general point (with reference to a Cartesian coordinate system). (a) At the point (1,0,2), calculate the rate of change of pressure per unit distance in the direction of the vector i- 2j+ 2k . (b) At (1,0,2) , find all directions along which the rate of change of pressure per unit distance is zero. (Give the directions using vectors in the form ai + bj+ ck .) (c) At (1,0,2), find the direction along which the rate of increase in the pressure per unit distance is the greatest. What is the greatest rate of increase in the pressure per unit distance at (1,0,2) ? Solution. We need Vpvdr002) in all the parts. Vp=Pi+ ốp j+ = [6x+ 2x2 exp(-2 y)]i – 2x z² exp(-2y)j+2x² zexp(-2y)k =141 – 8j+ 4k at (x, y, z) = (1,0, 2) (a) The vector i– 2j+ 2k has magnitude 1? + (-2)² + 2² = 3. The required rate is given by 2 2, 38 (141 – 8j+ 4k) • j+=k 3

1. The air pressure in a particular room varies from point to point and is given by p(x, y, 2) = 3x + xỉ exp(-2 y), where (x, y, z) describes a general point (with reference to a Cartesian coordinate system). (a) At the point (1,0,2), calculate the rate of change of pressure per unit distance in the direction of the vector i- 2j+ 2k . (b) At (1,0,2) , find all directions along which the rate of change of pressure per unit distance is zero. (Give the directions using vectors in the form ai + bj+ ck .) (c) At (1,0,2), find the direction along which the rate of increase in the pressure per unit distance is the greatest. What is the greatest rate of increase in the pressure per unit distance at (1,0,2) ? Solution. We need Vpvdr002) in all the parts. Vp=Pi+ ốp j+ = [6x+ 2x2 exp(-2 y)]i – 2x z² exp(-2y)j+2x² zexp(-2y)k =141 – 8j+ 4k at (x, y, z) = (1,0, 2) (a) The vector i– 2j+ 2k has magnitude 1? + (-2)² + 2² = 3. The required rate is given by 2 2, 38 (141 – 8j+ 4k) • j+=k 3

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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