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əz * əy? ax2 ar? r ar 32. Show that the Laplacian operator a/Əx² + a²/ay² + a /az? in three dimensions is given in cylindrical co- ordinates by the formula 1 3 ax2 ay? ar r ar

əz * əy? ax2 ar? r ar 32. Show that the Laplacian operator a/Əx² + a²/ay² + a /az? in three dimensions is given in cylindrical co- ordinates by the formula 1 3 ax2 ay? ar r ar

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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#32

ax² ar2 32. Show that the Laplacian operator a/ax² + a²/ay² + a2/az2 in three dimensions is given in cylindrical co- ordinates by the formula a 1 a a2 + az? 1 a2 ax2 ar? rar 33. In this problem, you will determine the formula for the Laplacian operator in spherical coordinates. (a) First, note that the cylindrical/spherical conver- sions given by formula (6) of §1.7 express the cylindrical coordinates z and r in terms of the spherical coordinates p and o by equations of pre- cisely the same form as those that express x and in terms of the polar coordinates r and 0. Use (b 37. Fine the y this fact to write a/ar in terms of a/ap and a/ay. (Also see formula (10) of this section.) (b) Use the ideas and result of part (a) to establish the following formula:
Transcribed Image Text:ax² ar2 32. Show that the Laplacian operator a/ax² + a²/ay² + a2/az2 in three dimensions is given in cylindrical co- ordinates by the formula a 1 a a2 + az? 1 a2 ax2 ar? rar 33. In this problem, you will determine the formula for the Laplacian operator in spherical coordinates. (a) First, note that the cylindrical/spherical conver- sions given by formula (6) of §1.7 express the cylindrical coordinates z and r in terms of the spherical coordinates p and o by equations of pre- cisely the same form as those that express x and in terms of the polar coordinates r and 0. Use (b 37. Fine the y this fact to write a/ar in terms of a/ap and a/ay. (Also see formula (10) of this section.) (b) Use the ideas and result of part (a) to establish the following formula:
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