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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

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.4: Spanning Sets And Linear Independence
Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...
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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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