Question

求解一道偏微分方程求解，谢谢，急急急

use the chain rule to find the indicated partial derivatives
N=(p+q)/(p+r), p=u+vw q=v+uw r=w+uv ;
partial N/partial u , partial N/partial v , partial N/ partial w , when u=2 ,v=3 , w=4 谢谢

Answer 1

when u=2 ,v=3 , w=4 then p=14 q=11 r=10
∵∂N/∂p=1/(p+r)-(p+q)/(p+r)^2,
∂N/∂q=1/(p+r),
∂N/∂r=-(p+q)/(p+r)^2
∂p/∂u=1 ∂q/∂u=w ∂r/∂u=v
∂p/∂v=w ∂q/∂v=1 ∂r/∂v=u
∂p/∂w=v ∂q/∂w=u ∂r/∂w=1
∴when u=2 ,v=3 , w=4 and p=14 q=11 r=10
∂N/∂p=-1/576 ∂N/∂q=1/24 ∂N/∂r=-25/576

∂p/∂u=1 ∂q/∂u=4 ∂r/∂u=3
∂p/∂v=4 ∂q/∂v=1 ∂r/∂v=2
∂p/∂w=3 ∂q/∂w=2 ∂r/∂w=1
there for
∂N/∂u=(∂N/∂p)(∂p/∂u)+(∂N/∂q)(∂q/∂u)+(∂N/∂r)(∂r/∂u)=(-1/576)+(1/24)4+(-25/576)3=5/144
∂N/∂v=(∂N/∂p)(∂p/∂v)+(∂N/∂q)(∂q/∂v)+(∂N/∂r)(∂r/∂v)=(-1/576)4+(1/24)+(-25/5762=-15/288
∂N/∂w=(∂N/∂p)(∂p/∂w)+(∂N/∂q)(∂q/∂w)+(∂N/∂r)(∂r/∂w)=(-1/576)3+(1/24)2+(-25/576)=5/144

Answer 2

N=(u+v+vw+uw)/(u+w+vw+uv)=(u+v)(1+w)/[(u+w)(1+v)]
N'u=(1+w)/(1+v)*[(u+w)-(u+v)]/(u+w)^2=(1+w)/(1+v)*(w-v)/(u+w)^2
N'v=(1+w)/(u+w)*[(1+v)-(u+v)]/(1+v)^2=(1+w)/(u+w)*(1-u)/(1+v)^2
N'w=(u+v)/(1+v)*[(u+w)-(1+w)]/(u+w)^2=(u+v)/(1+v)*(u-1)/(u+w)^2
当u=2, v=3,w=4时，代入得：
N'u=5/4*1/6^2=5/144
N'v=5/6*(-1)/4^2=-5/96
N'w=5/4*1/6^2=5/144