Question

Statement-I: The function \( u = x^2 + y^2 \), \( v = \tan^{-1}\left(\frac{y}{x}\right) \) are functionally independent.
Statement-II: The Jacobian \( \frac{\partial(u,v){\partial(x,y)} \) is non-zero.
The correct answer is}

• A. Statement-I is true, but statement-II is false
• B. Statement-I is false, but statement-II is true
• C. Both statement-I and statement-II are true
• D. Both statement-I and statement-II are false

Hint:

Notice that \( u = r^2 \) and \( v = \theta \) in polar coordinates! Since polar coordinates map points uniquely to independent coordinate axes, functions of \( r \) alone and \( \theta \) alone are always functionally independent.

Answer 1

The Correct Option is C