Question

高数一道方程组求偏导，有图，谢谢

Answer 1

x = u+v, 两边对 x 求偏导， 1 = ∂u/∂x + ∂v/∂x， (1)
两边对 y 求偏导， 0 = ∂u/∂y + ∂v/∂y，(2)
y = u^2+v^2, 两边对 x 求偏导， 0 = 2u∂u/∂x + 2v∂v/∂x，(3)
两边对 y 求偏导， 1 = 2u∂u/∂y + 2v∂v/∂y，(4)
(1)(3) 联立解得 ∂u/∂x = -v/(u-v), ∂v/∂x = u/(u-v);
(2)(4) 联立解得 ∂u/∂y = 1/[2(u-v)], ∂v/∂u = -1/[2(u-v)].
z = u^3+v^3
∂z/∂x = 3u^2∂u/∂x + 3v^2∂v/∂x = 3(-vu^2+uv^2)/(u-v) = -3uv
∂z/∂y = 3u^2∂u/∂y + 3v^2∂v/∂y = 3(u^2-v^2)/[2(u-v)] = (3/2)(u+v)

Answer 2

蓟废黜咨w