Question

Statement-I: 
The functions \[ u=x^2+y^2,\qquad v=\tan^{-1}\left(\frac{y}{x}\right) \] 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