如何解决这道向量的题目

Question

s1=2，s2=0，s3=0. Find a vector field F~ : R 3 → R 3 that has ALL of the following four properties:
(i) F~ is the curl of a certain vector field,
(ii) F~ is the gradient of a certain scalar field.
(iii) Each term of F~ is a function of x, y and z and cannot be a constant,i.e., the form of the vector field: F~ = f1(x, y, z) i + f2(x, y, z) ~j + f3(x, y, z) ~k.
(iv) F = (s1 + 3) ~i + (s2 + 2) ~j + (s3 + 1) ~k, when (x, y, z) = (s1, s2, s3).
Explain your reasoning.

Answer 1

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