Computation_of_partial_derivatives

Computation_of_partial_derivatives - ABOUT THE RELATIONS OF...

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ABOUT THE RELATIONS OF THE PARTIAL DERIVATIVES RELATED TO ( x,y ) AND ( r,θ ) Suppose f ( x,y ) has continuous second order partial derivatives. Under the polar transformation x = r cos θ , y = r sin θ . Show that (a)( ∂f ∂x ) 2 + ( ∂f ∂y ) 2 = ( ∂f ∂r ) 2 + 1 r 2 ( ∂f ∂θ ) 2 (b) 2 f ∂x 2 + 2 f ∂y 2 = 2 f ∂r 2 + 1 r 2 2 f ∂θ 2 + 1 r ∂f ∂r . Proof. ∂f ∂r = ∂f ∂x ∂x ∂r + ∂f ∂y ∂y ∂r = ∂f ∂x cos θ + ∂f ∂y sin θ 2 f ∂r 2 = ∂r ( ∂f ∂r ) = ∂r ( ∂f ∂x cos θ + ∂f ∂y sin θ ) = ∂r ( ∂f ∂x ) cos θ + ∂r ( ∂f ∂y ) sin θ ∂r ( ∂f ∂x ) = ∂x ( ∂f ∂x ) ∂x ∂r + ∂y ( ∂f ∂x ) ∂y ∂r = 2 f ∂x 2 cos θ + 2 f ∂y∂x sin θ ∂r ( ∂f ∂y ) = ∂x ( ∂f ∂y ) ∂x ∂r + ∂y ( ∂f ∂y ) ∂y ∂r = 2 f ∂x∂y cos θ + 2 f ∂y 2 sin θ Hence 2 f ∂r 2 = [ 2 f ∂x 2 cos θ + 2 f ∂y∂x sin
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Computation_of_partial_derivatives - ABOUT THE RELATIONS OF...

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