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3(e). If it is true that the line integral F. dr 0 for all closed curves C₁ show that the line integral over a non-closed curve depends only on the beginning- and end-points of the curve.

3(e). If it is true that the line integral F. dr 0 for all closed curves C₁ show that the line integral over a non-closed curve depends only on the beginning- and end-points of the curve.

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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Page 14 of 27 3(c). If it is true that the line integral F F. dr 0 for all closed curves C, = show that the line integral over a non-closed curve depends only on the beginning- and end-points of the curve. 3(d). Suppose that F is given by F(u, v, w) = ueu + we, + v² ew. with respect to an orthogonal curvilinear coordinate system (u, v, w) with scale factors h = w, h = u and he v. Determine div F. Multivariable and Complex Calculus II
Transcribed Image Text:Page 14 of 27 3(c). If it is true that the line integral F F. dr 0 for all closed curves C, = show that the line integral over a non-closed curve depends only on the beginning- and end-points of the curve. 3(d). Suppose that F is given by F(u, v, w) = ueu + we, + v² ew. with respect to an orthogonal curvilinear coordinate system (u, v, w) with scale factors h = w, h = u and he v. Determine div F. Multivariable and Complex Calculus II
Multivariable and Complex Calculus II Question 3. Let F: R³ R³ be any C² vector field. → 3(a). Prove that the divergence of the curl of F is zero. 3(b). For F as defined above, a misguided professor claims that for any closed curve C, F. dr = 0 because of the argument: $ F-dr - 11.S ( ( ▼ × F)-dS = [], div (curl F) dV = [] oav = 0, by using Stokes theorem, the divergence theorem, and then part (a), for an appropriately chosen surface S and volume W. Carefully explain all the errors in this argument. Page 12 of 27
Transcribed Image Text:Multivariable and Complex Calculus II Question 3. Let F: R³ R³ be any C² vector field. → 3(a). Prove that the divergence of the curl of F is zero. 3(b). For F as defined above, a misguided professor claims that for any closed curve C, F. dr = 0 because of the argument: $ F-dr - 11.S ( ( ▼ × F)-dS = [], div (curl F) dV = [] oav = 0, by using Stokes theorem, the divergence theorem, and then part (a), for an appropriately chosen surface S and volume W. Carefully explain all the errors in this argument. Page 12 of 27
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