设对于任意 x 和 y，有 ((partial f)/(partial x))^2 + ((partial f)/(partial y))^2 = 4， 用变量替换 {x=uv, y=(1)/(2)(u^2-v^2). 将 f(x,y) 变换成函数 g(u,v)，试求满足关系式 a((partial g)/(partial u))^2 - b((partial g)/(partial v))^2 = u^2 + v^2 中的常数 a = ( ), b = ( ).A. a = (1)/(4), b = -(1)/(4)；B. a = (1)/(3), b = -(1)/(3)；C. a = (1)/(2), b = -(1)/(2)；D. a = 1, b = -1.

设对于任意 $x$ 和 $y$，有 $\left(\frac{\partial f}{\partial x}\right)^2 + \left(\frac{\partial f}{\partial y}\right)^2 = 4$， 用变量替换 $\left\{\begin{array}{l}x=uv, \\ y=\frac{1}{2}(u^2-v^2)\end{array}\right.$ 将 $f(x,y)$ 变换成函数 $g(u,v)$，试求满足关系式 $a\left(\frac{\partial g}{\partial u}\right)^2 - b\left(\frac{\partial g}{\partial v}\right)^2 = u^2 + v^2$ 中的常数 $a = (\ )$, $b = (\ )$.

A. $a = \frac{1}{4}, b = -\frac{1}{4}$；

B. $a = \frac{1}{3}, b = -\frac{1}{3}$；

C. $a = \frac{1}{2}, b = -\frac{1}{2}$；

D. $a = 1, b = -1$.