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

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.6: Additional Trigonometric Graphs

Problem 78E

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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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