(a) Let D = R² \ {(0,0)} and consider the vector field f defined by: f(x, y) = f(x, y)i + f2(x, y) j where X x² + y² x³ x²y = + +2xy cos(x²y) and f2(x, y) 6 2 fi(x, y) Show that f is a gradient field by finding a potential function a scalar function o such that Vo = f. + 23 + Y x² + y² 6 for it; i.e, find -+x² cos(x-

(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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