4. Let V : R? → R be a differentiable function. (i) Suppose r : R → R? satisfies the equation (t) = -VV(x(t)) for all teR. Prove that the function V(r(t)) is non-increasing. (ii) Suppose r : R → R² satisfies the equation 0 1 r(t) = JVV(x(t)), J = for all t e R. Prove that the function V(r(t)) is constant.

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Q: If V=R\power{3}, and W\index{1} is the xy plane and  let W\index{2} is yz-plane:…

A: Let (a,b,c) be any element of R3, then (a,b,c)=(x,y,0)+(0,y,z)=(x,2y,z) ⇒a=x, b=2y, c=z ⇒a,b,c∈R ⇒…