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(a) If F is a C¹ vector field on R³ with V. F= div(F) = 0, then F = Vf for some scalar function f. True False (b) If F is a C¹ vector field on R³ with V. F = div(F) = 0, then F = V x G for some vector field G. True False (c) If F is a C¹ vector field on R³ - {(0, 0, 0)} (i.e. all points in three-space except the origin) with V. F = div(F) = 0, then F= V x G for some vector field G. True False (d) If F is a C¹ vector field on R³ - {(0, 0, 0)} (i.e. all points in three-space except the origin) with V x F = curl(F) = 0, then F = Vf for some scalar function f. True False

(a) If F is a C¹ vector field on R³ with V. F= div(F) = 0, then F = Vf for some scalar function f. True False (b) If F is a C¹ vector field on R³ with V. F = div(F) = 0, then F = V x G for some vector field G. True False (c) If F is a C¹ vector field on R³ - {(0, 0, 0)} (i.e. all points in three-space except the origin) with V. F = div(F) = 0, then F= V x G for some vector field G. True False (d) If F is a C¹ vector field on R³ - {(0, 0, 0)} (i.e. all points in three-space except the origin) with V x F = curl(F) = 0, then F = Vf for some scalar function f. True False

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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(a) If F is a C¹ vector field on R³ with V. F = div(F) = 0, then F = Vƒ for some scalar function f. True False (b) If F is a C¹ vector field on R³ with V · F = div(F) = 0, then F = V x G for some vector field G. True False (c) If F is a C¹ vector field on R³ {(0, 0, 0)} (i.e. all points in three-space except the origin) with V. F = div(F) = 0, then F= V x G for some vector field G. True False (d) If F is a C¹ vector field on R³ - {(0, 0, 0)} (i.e. all points in three-space except the origin) with V × F = curl(F) = 0, Vf for some scalar function f. then F = True False
Transcribed Image Text:(a) If F is a C¹ vector field on R³ with V. F = div(F) = 0, then F = Vƒ for some scalar function f. True False (b) If F is a C¹ vector field on R³ with V · F = div(F) = 0, then F = V x G for some vector field G. True False (c) If F is a C¹ vector field on R³ {(0, 0, 0)} (i.e. all points in three-space except the origin) with V. F = div(F) = 0, then F= V x G for some vector field G. True False (d) If F is a C¹ vector field on R³ - {(0, 0, 0)} (i.e. all points in three-space except the origin) with V × F = curl(F) = 0, Vf for some scalar function f. then F = True False
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