Answer: Let Prove or disprove that there is a vector-valued function with the following

Chapter 15, Problem 25

(choose chapter or problem)

Let \(\mathbf{G}(x, y)=\left(\frac{-y}{x^{2}+4 y^{2}}, \frac{x}{x^{2}+4 y^{2}}, 0\right)\).

Prove or disprove that there is a vector-valued function F(x, y, z) = (M(x, y, z), N(x, y, z), P(x, y, z)) with the following properties:

(i) M, N, P have continuous partial derivatives for all \((x, y, z) \neq(0,0,0)\);

(ii) Curl F= 0 for all \((x, y, z) \neq(0,0,0)\);

(iii) F(x, y, 0) = G(x, y).

Text Transcription:

G(x, y) = (-y / x^2 + 4y^2, x / x^2 + 4y^2, 0)

(x, y, z) neq (0,0,0)