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(a) Let F= 2xz2i+j+xy³zk and f(x, y, z) = x²y. Compute the following quantities: (i) Vf (ii) curl(F) (iii) F x Vf (iv) F. Vf (b) Does there exist a vector field G such that curl(G) = (xsin y, cos y, z - xy)? (c) Suppose that F, G: R³ R³ are C¹ vector filelds and f,g: R³ Rare C2 real-valued functions. Prove the following statements: (i) div(F x G) = G curl(F) - F-curl (G) (ii) div(Vfx Vg) = 0

(a) Let F= 2xz2i+j+xy³zk and f(x, y, z) = x²y. Compute the following quantities: (i) Vf (ii) curl(F) (iii) F x Vf (iv) F. Vf (b) Does there exist a vector field G such that curl(G) = (xsin y, cos y, z - xy)? (c) Suppose that F, G: R³ R³ are C¹ vector filelds and f,g: R³ Rare C2 real-valued functions. Prove the following statements: (i) div(F x G) = G curl(F) - F-curl (G) (ii) div(Vfx Vg) = 0

Algebra & Trigonometry with Analytic Geometry
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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(a) Let F = 2xz²i+j+xy³zk and f(x, y, z) = x²y. Compute the following quantities: (i) Vf (ii) curl(F) (iii) F x Vf (b) Does there exist a vector field G such that curl(G) = (xsin y, cos y, z - xy)? (c) Suppose that F, G : R³ R³ are C¹ vector filelds and f,g: R³ → Rare C² real-valued functions. Prove the following statements: (i) div(F x G) = G curl (F) - F-curl (G) (ii) div(Vfx Vg) = 0 (iv) F. Vf
Transcribed Image Text:(a) Let F = 2xz²i+j+xy³zk and f(x, y, z) = x²y. Compute the following quantities: (i) Vf (ii) curl(F) (iii) F x Vf (b) Does there exist a vector field G such that curl(G) = (xsin y, cos y, z - xy)? (c) Suppose that F, G : R³ R³ are C¹ vector filelds and f,g: R³ → Rare C² real-valued functions. Prove the following statements: (i) div(F x G) = G curl (F) - F-curl (G) (ii) div(Vfx Vg) = 0 (iv) F. Vf
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