prove that (du/dr)^2+1/r^2(du/dz)^2=(du/dx)^2+(du/dy)^2
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Step-by-step explanation:
Answer
r
2
=x
2
+y
2
+z
2
,v=f(r)
∂x
∂U
=
∂r
∂U
⋅
∂x
∂r
=f
′
(r)
r
x
similarly
∂y
∂U
=f
′
(r)
π
y
∂z
∂U
=f
′
(π)
r
z
∂x
2
∂
2
U
=
∂x
∂
(f(r)
r
x
)
=
r
x
f
′′
(r)
π
x
+f
′
(r)(
r
1
−
r
2
x
∂x
∂r
)
=f
′′
(r)
r
2
x
2
+f
′
(r)(
r
1
−
r
3
x
2
)
∂y
2
∂
2
v
=f
′′
(r)
r
2
x
2
+f
′
(r)(
r
1
−
r
3
y
2
)
∂z
2
∂
2
v
=f
′′
(r)
r
2
x
2
+f
′
(r)(
r
1
−
r
3
z
2
)
thus RHS=f
′′
(r)+f
′
(r)
r
2
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