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Necessary and sufficient condition for a Jacobian to be zero

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Real Analysis
Jacobians(I)
Necessary and sufficient condition for the value of a Jacobian of n independent functions to be zero

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Necessary and sufficient condition for the value of a Jacobian to be zero is explained in detailed.
It also explains for the existence of a relation between n independent functions. The solution is detailed and well presented.

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Definition:- If are the functions of variables then the determinant

is called the Jacobian of with respect to . It is usually denoted either by
or, or, .

Theorem:- Let be the functions of independent variables .
Then there exist a relation if and only if
i,e., .

Solution:- Suppose the relation
...

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