Answered: Determine the value of k for which (u,…

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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Question

7. (a) Determine the value of k for which (u, v, w) is an orthogonal coordinate system if
x = −(u² + kv²), y = uv and z = w.
(b) Given that F and G are vector fields with G a vector potential of F, prove that G is not
unique.
(c) Show that a vector field F =
(4uv − θ³/√u² + v², 2u²/√u² + v², (lnθ − 3uθ²/uv), defined in paraboloidal
coordinate system (x = uv cos θ, y = uv sin θ, z =1/2(u² − v²)
is irrotational and hence find its scalar potential.

System that uses coordinates to uniquely determine the position of points. The most common coordinate system is the Cartesian system, where points are given by distance along a horizontal x-axis and vertical y-axis from the origin. A polar coordinate system locates a point by its direction relative to a reference direction and its distance from a given point. In three dimensions, it leads to cylindrical and spherical coordinates.

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