Solution;
(a)
Given;
u=xy
v=y
w=x+z
Now;
x=yu=vu
y=v
z=w−x=w−vu
∂(u,v,w)∂(x,y,z)=∣∣∂u∂x∂u∂y∂u∂z∂v∂x∂v∂y∂v∂z∂w∂x∂w∂y∂w∂z∣∣ =∣∣v10−v1v2u1−v2u001∣∣ =v1(1−0)−v2u(0−0)+0(0+v1)
Hence;
∂(u,v,w)∂(x,y,z)=v1
(b)
u=x+y+z
v=x+y-z
w=x-y+z
Now;
v+w=x+y−z+x−y+z=2x
Hence;
x=2v+w
Also;
u−v=x+y+z−x−y+z=2z
Hence;
z=2u−v
Also;
y=x+z−w =2v+w+2u−v−w
Hence;
y=2u−w
Therefore;
∂(u,v,w)∂(x,y,z)=∣∣∂u∂x∂u∂y∂u∂z∂v∂x∂v∂y∂v∂z∂w∂x∂w∂y∂w∂z∣∣=∣∣02121210−2121−210∣∣=0(0+41)−21(0+41)+21(−41−0)
Therefore;
∂(u,v,w)∂(x,y,z)=−41
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