Prove the converse of Exercise 29 as follows: Let u D f .x;
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Prove the converse of Exercise 29 as follows: Let u D f .x; y/ and v D g.x; y/, and suppose that @.u; v/=@.x; y/ D @.f;
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Prove the converse of Exercise 29 as follows: Let u D f .x; y/ and v D g.x; y/, and suppose that @.u; v/=@.x; y/ D @.f; g/=@.x; y/ is identically zero for all x and y. Show that .@u=@x/v is identically zero. Hence u, considered as a function of x and v, is independent of x; that is, u D k.v/ for some function k of one variable. Why does this imply that f and g are functionally dependent?
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