Discuss sufficient condition for the value of a jacobian

Question

Necessary and sufficient condition for a Jacobian to be zero

Response to the following:

Real Analysis

Jacobians(I)

Necessary and sufficient condition for the value of a Jacobian of n independent functions to be zero

Theorem:- Let ƒ₁, ƒ₂,....,ƒ_n be the functions of independent variables x₁,x₂,...,x_n

Then there exist a relation F( ƒ₁, ƒ₂,...,ƒ_n ) =0 if and only if J(( ƒ₁, ƒ₂,....,ƒ_n ) =0

i,e., ∂ (ƒ₁, ƒ₂,....,ƒ_n)/ ∂ (x₁,x₂,....,x_n)=0