Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 72886 by mhmd last updated on 04/Nov/19

let w=f(x, y) be a differentiable function where x=rcosθ and y=rsinθ show that (f_x )^2 +(f_y )^2 =(w_x )^2 +1/r^2 (w_y )^2 ? help me sir

letw=f(x,y)beadifferentiablefunctionwherex=rcosθandy=rsinθshowthat(fx)2+(fy)2=(wx)2+1/r2(wy)2?helpmesir

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

w(r,θ)=f(rcos(θ),rsin(θ)) =f(B(r,θ) ⇒((∂w/∂r),(∂w/∂θ))=((∂f/∂x),(∂f/∂y)). (((cos(θ) −rsin(θ))),((sin(θ) rcos(θ))) ) ⇒((∂f/∂x),(∂f/∂y))=((∂w/∂r),(∂w/∂θ)). (((cos(θ) −rsin(θ))),((sin(θ) rcos(θ))) )^(−1) =((∂w/∂r),(∂w/∂θ)).(1/r). (((rcos(θ) rsin(θ))),((−sin(θ) cos(θ) )) ) (∂f/∂x)=cos(θ)(∂w/∂r)−((sin(θ))/r).(∂w/∂θ) (∂f/∂y)=sin(θ)(∂w/∂r)+((cos(θ))/r).(∂w/∂θ) ⇒((∂f/∂x))^2 +((∂f/∂y))^2 =(cos(θ)(∂w/∂r)−((sin(θ))/r)(∂w/∂θ))^2 +(sin(θ)(∂w/∂r)+((cos(θ))/r)(∂w/∂θ))^2 =((∂w/∂r))^2 +(1/r^2 ).((∂w/∂θ))^2

w(r,θ)=f(rcos(θ),rsin(θ))=f(B(r,θ)⇒(∂w∂r,∂w∂θ)=(∂f∂x,∂f∂y).(cos(θ)−rsin(θ)sin(θ)rcos(θ))⇒(∂f∂x,∂f∂y)=(∂w∂r,∂w∂θ).(cos(θ)−rsin(θ)sin(θ)rcos(θ))−1=(∂w∂r,∂w∂θ).1r.(rcos(θ)rsin(θ)−sin(θ)cos(θ))∂f∂x=cos(θ)∂w∂r−sin(θ)r.∂w∂θ∂f∂y=sin(θ)∂w∂r+cos(θ)r.∂w∂θ⇒(∂f∂x)2+(∂f∂y)2=(cos(θ)∂w∂r−sin(θ)r∂w∂θ)2+(sin(θ)∂w∂r+cos(θ)r∂w∂θ)2=(∂w∂r)2+1r2.(∂w∂θ)2

Terms of Service

Privacy Policy

Contact: info@tinkutara.com