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Question Number 72883 by mhmd last updated on 04/Nov/19

if w=f(u and v) where f_(uu) +f_(vv) =0 and u=(x^2 −y^2 )/2 and v=xy show that w_(xx) +w_(yy) =0 ? pleas sir help me

ifw=f(uandv)wherefuu+fvv=0andu=(x2−y2)/2andv=xyshowthatwxx+wyy=0?pleassirhelpme

Answered by mind is power last updated on 04/Nov/19

w(x,y)=f(u,v) w(x,y)=f(β(x,y)) ⇒jac W=jac(f).jacβ(x,y) ⇒((∂w/∂x),(∂w/∂y))=((∂f/∂u),(∂f/∂v)). [((x −y)),((y x)) ] ⇒(∂w/∂x)=((x∂f)/∂u)+((y∂f)/∂v) ⇒(∂w/∂y)=−y(∂f/∂u)+x(∂f/∂v) (∂^2 W/∂x^2 )=(∂f/∂u)+x((∂^2 f/∂u^2 ).x+(∂^2 f/(∂u∂v))y)+y((∂^2 f/(∂v∂u))x+(∂^2 f/∂v^2 )y) (∂^2 w/∂y^2 )=−(∂f/∂u)−y((∂^2 f/∂u^2 ).−y+(∂^2 f/(∂u∂v))x)+x((∂^2 f/(∂u∂v)).−y+(∂^2 f/∂v^2 ).x) ⇒(∂^2 w/∂x^2 )=(∂f/∂u)+2xy(∂^2 f/(∂u∂v))+x^2 (∂^2 f/∂u^2 )+y^2 (∂^2 f/∂v^2 ) (∂^2 w/∂y^2 )=−(∂f/∂u)−2xy(∂^2 f/(∂u∂v))+x^2 (∂^2 f/∂v^2 )+y^2 (∂^2 f/∂u^2 ) ⇒(∂^2 w/∂x^2 )+(∂^2 w/∂y^2 )=(x^2 +y^2 )[(∂^2 f/∂u^2 )+(∂^2 f/∂v^2 )]=(x^2 +y^2 ).0=0

w(x,y)=f(u,v)w(x,y)=f(β(x,y))⇒jacW=jac(f).jacβ(x,y)⇒(∂w∂x,∂w∂y)=(∂f∂u,∂f∂v).[x−yyx]⇒∂w∂x=x∂f∂u+y∂f∂v⇒∂w∂y=−y∂f∂u+x∂f∂v∂2W∂x2=∂f∂u+x(∂2f∂u2.x+∂2f∂u∂vy)+y(∂2f∂v∂ux+∂2f∂v2y)∂2w∂y2=−∂f∂u−y(∂2f∂u2.−y+∂2f∂u∂vx)+x(∂2f∂u∂v.−y+∂2f∂v2.x)⇒∂2w∂x2=∂f∂u+2xy∂2f∂u∂v+x2∂2f∂u2+y2∂2f∂v2∂2w∂y2=−∂f∂u−2xy∂2f∂u∂v+x2∂2f∂v2+y2∂2f∂u2⇒∂2w∂x2+∂2w∂y2=(x2+y2)[∂2f∂u2+∂2f∂v2]=(x2+y2).0=0

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