For u ( x , y ) {\displaystyle u(x,y)} where x = x ( s , t ) {\displaystyle x=x(s,t)} , y = y ( s , t ) {\displaystyle y=y(s,t)} : ∂ 2 u ∂ s 2 = ∂ 2 u ∂ x 2 ( ∂ x ∂ s ) 2 + ∂ 2 u ∂ x ∂ y ∂ x ∂ s ∂ y ∂ s + ∂ u ∂ x ∂ 2 x ∂ s 2 + ∂ 2 u ∂ y ∂ x ∂ x ∂ s ∂ y ∂ s + ∂ 2 u ∂ y 2 ( ∂ y ∂ s ) 2 + ∂ u ∂ y ∂ 2 y ∂ s 2 {\displaystyle {\frac {\partial ^{2}u}{\partial s^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}\left({\frac {\partial x}{\partial s}}\right)^{2}+{\frac {\partial ^{2}u}{\partial x\partial y}}{\frac {\partial x}{\partial s}}{\frac {\partial y}{\partial s}}+{\frac {\partial u}{\partial x}}{\frac {\partial ^{2}x}{\partial s^{2}}}+{\frac {\partial ^{2}u}{\partial y\partial x}}{\frac {\partial x}{\partial s}}{\frac {\partial y}{\partial s}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\left({\frac {\partial y}{\partial s}}\right)^{2}+{\frac {\partial u}{\partial y}}{\frac {\partial ^{2}y}{\partial s^{2}}}} ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2 ( ∂ x ∂ t ) 2 + ∂ 2 u ∂ x ∂ y ∂ x ∂ t ∂ y ∂ t + ∂ u ∂ x ∂ 2 x ∂ t 2 + ∂ 2 u ∂ y ∂ x ∂ x ∂ t ∂ y ∂ t + ∂ 2 u ∂ y 2 ( ∂ y ∂ t ) 2 + ∂ u ∂ y ∂ 2 y ∂ t 2 {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}\left({\frac {\partial x}{\partial t}}\right)^{2}+{\frac {\partial ^{2}u}{\partial x\partial y}}{\frac {\partial x}{\partial t}}{\frac {\partial y}{\partial t}}+{\frac {\partial u}{\partial x}}{\frac {\partial ^{2}x}{\partial t^{2}}}+{\frac {\partial ^{2}u}{\partial y\partial x}}{\frac {\partial x}{\partial t}}{\frac {\partial y}{\partial t}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\left({\frac {\partial y}{\partial t}}\right)^{2}+{\frac {\partial u}{\partial y}}{\frac {\partial ^{2}y}{\partial t^{2}}}}
This is why it's called calc-u-lose. D:
Let u = f ( x , y ) {\displaystyle u=f(x,y)} , x = e s cos t {\displaystyle x=e^{s}\cos t} , y = e s sin t {\displaystyle y=e^{s}\sin t} . Show that e − 2 s ( ∂ 2 u ∂ s 2 + ∂ 2 u ∂ t 2 ) = ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 {\displaystyle e^{-2s}\left({\frac {\partial ^{2}u}{\partial s^{2}}}+{\frac {\partial ^{2}u}{\partial t^{2}}}\right)={\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}} .
∂ x ∂ s = e s cos t {\displaystyle {\frac {\partial x}{\partial s}}=e^{s}\cos t} ∂ x ∂ t = − e s sin t {\displaystyle {\frac {\partial x}{\partial t}}=-e^{s}\sin t}
∂ 2 x ∂ s 2 = e s cos t {\displaystyle {\frac {\partial ^{2}x}{\partial s^{2}}}=e^{s}\cos t} ∂ 2 x ∂ t 2 = − e s cos t {\displaystyle {\frac {\partial ^{2}x}{\partial t^{2}}}=-e^{s}\cos t}
∂ y ∂ s = e s sin t {\displaystyle {\frac {\partial y}{\partial s}}=e^{s}\sin t} ∂ y ∂ t = e s cos t {\displaystyle {\frac {\partial y}{\partial t}}=e^{s}\cos t}
∂ 2 y ∂ s 2 = e s sin t {\displaystyle {\frac {\partial ^{2}y}{\partial s^{2}}}=e^{s}\sin t} ∂ 2 y ∂ t 2 = − e s sin t {\displaystyle {\frac {\partial ^{2}y}{\partial t^{2}}}=-e^{s}\sin t}
∂ u ∂ s = ∂ u ∂ x ∂ x ∂ s + ∂ u ∂ y ∂ y ∂ s {\displaystyle {\frac {\partial u}{\partial s}}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial s}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial s}}} ∂ u ∂ t = ∂ u ∂ x ∂ x ∂ t + ∂ u ∂ y ∂ y ∂ t {\displaystyle {\frac {\partial u}{\partial t}}={\frac {\partial u}{\partial x}}{\frac {\partial x}{\partial t}}+{\frac {\partial u}{\partial y}}{\frac {\partial y}{\partial t}}}
∂ 2 u ∂ s 2 = ∂ ∂ s ( ∂ u ∂ x ) ⋅ ∂ x ∂ s + ∂ u ∂ x ∂ 2 x ∂ s 2 + ∂ ∂ s ( ∂ u ∂ y ) ⋅ ∂ y ∂ s + ∂ u ∂ y ∂ 2 y ∂ s 2 {\displaystyle {\frac {\partial ^{2}u}{\partial s^{2}}}={\frac {\partial }{\partial s}}\left({\frac {\partial u}{\partial x}}\right)\cdot {\frac {\partial x}{\partial s}}+{\frac {\partial u}{\partial x}}{\frac {\partial ^{2}x}{\partial s^{2}}}+{\frac {\partial }{\partial s}}\left({\frac {\partial u}{\partial y}}\right)\cdot {\frac {\partial y}{\partial s}}+{\frac {\partial u}{\partial y}}{\frac {\partial ^{2}y}{\partial s^{2}}}} ∂ 2 u ∂ t 2 = ∂ ∂ t ( ∂ u ∂ x ) ⋅ ∂ x ∂ t + ∂ u ∂ x ∂ 2 x ∂ t 2 + ∂ ∂ t ( ∂ u ∂ y ) ⋅ ∂ y ∂ t + ∂ u ∂ y ∂ 2 y ∂ t 2 {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial }{\partial t}}\left({\frac {\partial u}{\partial x}}\right)\cdot {\frac {\partial x}{\partial t}}+{\frac {\partial u}{\partial x}}{\frac {\partial ^{2}x}{\partial t^{2}}}+{\frac {\partial }{\partial t}}\left({\frac {\partial u}{\partial y}}\right)\cdot {\frac {\partial y}{\partial t}}+{\frac {\partial u}{\partial y}}{\frac {\partial ^{2}y}{\partial t^{2}}}}
∂ ∂ s ( ∂ u ∂ x ) = ∂ ∂ x ( ∂ u ∂ x ) ⋅ ∂ x ∂ s + ∂ ∂ x ( ∂ u ∂ y ) ⋅ ∂ y ∂ s = ∂ 2 u ∂ x 2 ∂ x ∂ s + ∂ 2 u ∂ x ∂ y ∂ y ∂ s {\displaystyle {\begin{aligned}{\frac {\partial }{\partial s}}\left({\frac {\partial u}{\partial x}}\right)&={\frac {\partial }{\partial x}}\left({\frac {\partial u}{\partial x}}\right)\cdot {\frac {\partial x}{\partial s}}+{\frac {\partial }{\partial x}}\left({\frac {\partial u}{\partial y}}\right)\cdot {\frac {\partial y}{\partial s}}\\&={\frac {\partial ^{2}u}{\partial x^{2}}}{\frac {\partial x}{\partial s}}+{\frac {\partial ^{2}u}{\partial x\partial y}}{\frac {\partial y}{\partial s}}\\\end{aligned}}} ∂ ∂ s ( ∂ u ∂ y ) = ∂ ∂ y ( ∂ u ∂ x ) ⋅ ∂ x ∂ s + ∂ ∂ y ( ∂ u ∂ y ) ⋅ ∂ y ∂ s = ∂ 2 u ∂ y ∂ x ∂ x ∂ s + ∂ 2 u ∂ y 2 ∂ y ∂ s {\displaystyle {\begin{aligned}{\frac {\partial }{\partial s}}\left({\frac {\partial u}{\partial y}}\right)&={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right)\cdot {\frac {\partial x}{\partial s}}+{\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial y}}\right)\cdot {\frac {\partial y}{\partial s}}\\&={\frac {\partial ^{2}u}{\partial y\partial x}}{\frac {\partial x}{\partial s}}+{\frac {\partial ^{2}u}{\partial y^{2}}}{\frac {\partial y}{\partial s}}\\\end{aligned}}}
∂ ∂ t ( ∂ u ∂ x ) = ∂ ∂ x ( ∂ u ∂ x ) ⋅ ∂ x ∂ t + ∂ ∂ x ( ∂ u ∂ y ) ⋅ ∂ y ∂ t = ∂ 2 u ∂ x 2 ∂ x ∂ t + ∂ 2 u ∂ x ∂ y ∂ y ∂ t {\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}\left({\frac {\partial u}{\partial x}}\right)&={\frac {\partial }{\partial x}}\left({\frac {\partial u}{\partial x}}\right)\cdot {\frac {\partial x}{\partial t}}+{\frac {\partial }{\partial x}}\left({\frac {\partial u}{\partial y}}\right)\cdot {\frac {\partial y}{\partial t}}\\&={\frac {\partial ^{2}u}{\partial x^{2}}}{\frac {\partial x}{\partial t}}+{\frac {\partial ^{2}u}{\partial x\partial y}}{\frac {\partial y}{\partial t}}\\\end{aligned}}} ∂ ∂ t ( ∂ u ∂ y ) = ∂ ∂ y ( ∂ u ∂ x ) ⋅ ∂ x ∂ t + ∂ ∂ y ( ∂ u ∂ y ) ⋅ ∂ y ∂ t = ∂ 2 u ∂ y ∂ x ∂ x ∂ t + ∂ 2 u ∂ y 2 ∂ y ∂ t {\displaystyle {\begin{aligned}{\frac {\partial }{\partial t}}\left({\frac {\partial u}{\partial y}}\right)&={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right)\cdot {\frac {\partial x}{\partial t}}+{\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial y}}\right)\cdot {\frac {\partial y}{\partial t}}\\&={\frac {\partial ^{2}u}{\partial y\partial x}}{\frac {\partial x}{\partial t}}+{\frac {\partial ^{2}u}{\partial y^{2}}}{\frac {\partial y}{\partial t}}\\\end{aligned}}}
∂ 2 u ∂ s 2 = ∂ 2 u ∂ x 2 ( ∂ x ∂ s ) 2 + ∂ 2 u ∂ x ∂ y ∂ x ∂ s ∂ y ∂ s + ∂ u ∂ x ∂ 2 x ∂ s 2 + ∂ 2 u ∂ y ∂ x ∂ x ∂ s ∂ y ∂ s + ∂ 2 u ∂ y 2 ( ∂ y ∂ s ) 2 + ∂ u ∂ y ∂ 2 y ∂ s 2 {\displaystyle {\frac {\partial ^{2}u}{\partial s^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}\left({\frac {\partial x}{\partial s}}\right)^{2}+{\frac {\partial ^{2}u}{\partial x\partial y}}{\frac {\partial x}{\partial s}}{\frac {\partial y}{\partial s}}+{\frac {\partial u}{\partial x}}{\frac {\partial ^{2}x}{\partial s^{2}}}+{\frac {\partial ^{2}u}{\partial y\partial x}}{\frac {\partial x}{\partial s}}{\frac {\partial y}{\partial s}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\left({\frac {\partial y}{\partial s}}\right)^{2}+{\frac {\partial u}{\partial y}}{\frac {\partial ^{2}y}{\partial s^{2}}}} ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2 ( ∂ x ∂ t ) 2 + ∂ 2 u ∂ x ∂ y ∂ x ∂ t ∂ y ∂ t + ∂ u ∂ x ∂ 2 x ∂ t 2 + ∂ 2 u ∂ y ∂ x ∂ x ∂ t ∂ y ∂ t + ∂ 2 u ∂ y 2 ( ∂ y ∂ t ) 2 + ∂ u ∂ y ∂ 2 y ∂ t 2 {\displaystyle {\frac {\partial ^{2}u}{\partial t^{2}}}={\frac {\partial ^{2}u}{\partial x^{2}}}\left({\frac {\partial x}{\partial t}}\right)^{2}+{\frac {\partial ^{2}u}{\partial x\partial y}}{\frac {\partial x}{\partial t}}{\frac {\partial y}{\partial t}}+{\frac {\partial u}{\partial x}}{\frac {\partial ^{2}x}{\partial t^{2}}}+{\frac {\partial ^{2}u}{\partial y\partial x}}{\frac {\partial x}{\partial t}}{\frac {\partial y}{\partial t}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\left({\frac {\partial y}{\partial t}}\right)^{2}+{\frac {\partial u}{\partial y}}{\frac {\partial ^{2}y}{\partial t^{2}}}}
∂ 2 u ∂ s 2 + ∂ 2 u ∂ t 2 = ∂ 2 u ∂ x 2 ( ∂ x ∂ s ) 2 + ∂ 2 u ∂ x ∂ y ∂ x ∂ s ∂ y ∂ s + ∂ u ∂ x ∂ 2 x ∂ s 2 + ∂ 2 u ∂ y ∂ x ∂ x ∂ s ∂ y ∂ s + ∂ 2 u ∂ y 2 ( ∂ y ∂ s ) 2 + ∂ u ∂ y ∂ 2 y ∂ s 2 + ∂ 2 u ∂ x 2 ( ∂ x ∂ t ) 2 + ∂ 2 u ∂ x ∂ y ∂ x ∂ t ∂ y ∂ t + ∂ u ∂ x ∂ 2 x ∂ t 2 + ∂ 2 u ∂ y ∂ x ∂ x ∂ t ∂ y ∂ t + ∂ 2 u ∂ y 2 ( ∂ y ∂ t ) 2 + ∂ u ∂ y ∂ 2 y ∂ t 2 = ∂ 2 u ∂ x 2 ( ( ∂ x ∂ s ) 2 + ( ∂ x ∂ t ) 2 ) + ∂ 2 u ∂ x ∂ y ( 2 ∂ x ∂ s ∂ y ∂ s + 2 ∂ x ∂ t ∂ y ∂ t ) + ∂ 2 u ∂ y 2 ( ( ∂ y ∂ s ) 2 + ( ∂ y ∂ t ) 2 ) + ∂ u ∂ x ( ∂ 2 x ∂ s 2 + ∂ 2 x ∂ t 2 ) + ∂ u ∂ y ( ∂ 2 y ∂ s 2 + ∂ 2 y ∂ t 2 ) = ∂ 2 u ∂ x 2 ( ( e s cos t ) 2 + ( − e s sin t ) 2 ) + ∂ 2 u ∂ x ∂ y ( 2 ( e s cos t ) ( e s sin t ) + 2 ( − e s sin t ) ( e s cos t ) ) + ∂ 2 u ∂ y 2 ( ( e s sin t ) 2 + ( e s cos t ) 2 ) + ∂ u ∂ x ( ( e s cos t ) + ( − e s cos t ) ) + ∂ u ∂ y ( ( e s sin t ) + ( − e s sin t ) ) = ∂ 2 u ∂ x 2 ( e 2 s cos 2 t + e 2 s sin 2 t ) + ∂ 2 u ∂ x ∂ y ( 2 e 2 s cos t sin t − 2 e 2 s sin t cos t ) + ∂ 2 u ∂ y 2 ( e 2 s sin 2 t + e 2 s cos 2 t ) + ∂ u ∂ x ( e s cos t − e s cos t ) + ∂ u ∂ y ( e s sin t − e s sin t ) = ∂ 2 u ∂ x 2 ( e 2 s ) + 0 + ∂ 2 u ∂ y 2 ( e 2 s ) + 0 + 0 = ( e 2 s ) ( ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 ) {\displaystyle {\begin{aligned}{\frac {\partial ^{2}u}{\partial s^{2}}}+{\frac {\partial ^{2}u}{\partial t^{2}}}&={\frac {\partial ^{2}u}{\partial x^{2}}}\left({\frac {\partial x}{\partial s}}\right)^{2}+{\frac {\partial ^{2}u}{\partial x\partial y}}{\frac {\partial x}{\partial s}}{\frac {\partial y}{\partial s}}+{\frac {\partial u}{\partial x}}{\frac {\partial ^{2}x}{\partial s^{2}}}+{\frac {\partial ^{2}u}{\partial y\partial x}}{\frac {\partial x}{\partial s}}{\frac {\partial y}{\partial s}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\left({\frac {\partial y}{\partial s}}\right)^{2}+{\frac {\partial u}{\partial y}}{\frac {\partial ^{2}y}{\partial s^{2}}}+{\frac {\partial ^{2}u}{\partial x^{2}}}\left({\frac {\partial x}{\partial t}}\right)^{2}+{\frac {\partial ^{2}u}{\partial x\partial y}}{\frac {\partial x}{\partial t}}{\frac {\partial y}{\partial t}}+{\frac {\partial u}{\partial x}}{\frac {\partial ^{2}x}{\partial t^{2}}}+{\frac {\partial ^{2}u}{\partial y\partial x}}{\frac {\partial x}{\partial t}}{\frac {\partial y}{\partial t}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\left({\frac {\partial y}{\partial t}}\right)^{2}+{\frac {\partial u}{\partial y}}{\frac {\partial ^{2}y}{\partial t^{2}}}\\&={\frac {\partial ^{2}u}{\partial x^{2}}}\left(\left({\frac {\partial x}{\partial s}}\right)^{2}+\left({\frac {\partial x}{\partial t}}\right)^{2}\right)+{\frac {\partial ^{2}u}{\partial x\partial y}}\left(2{\frac {\partial x}{\partial s}}{\frac {\partial y}{\partial s}}+2{\frac {\partial x}{\partial t}}{\frac {\partial y}{\partial t}}\right)+{\frac {\partial ^{2}u}{\partial y^{2}}}\left(\left({\frac {\partial y}{\partial s}}\right)^{2}+\left({\frac {\partial y}{\partial t}}\right)^{2}\right)+{\frac {\partial u}{\partial x}}\left({\frac {\partial ^{2}x}{\partial s^{2}}}+{\frac {\partial ^{2}x}{\partial t^{2}}}\right)+{\frac {\partial u}{\partial y}}\left({\frac {\partial ^{2}y}{\partial s^{2}}}+{\frac {\partial ^{2}y}{\partial t^{2}}}\right)\\&={\frac {\partial ^{2}u}{\partial x^{2}}}\left(\left(e^{s}\cos t\right)^{2}+\left(-e^{s}\sin t\right)^{2}\right)+{\frac {\partial ^{2}u}{\partial x\partial y}}\left(2\left(e^{s}\cos t\right)\left(e^{s}\sin t\right)+2\left(-e^{s}\sin t\right)\left(e^{s}\cos t\right)\right)+{\frac {\partial ^{2}u}{\partial y^{2}}}\left(\left(e^{s}\sin t\right)^{2}+\left(e^{s}\cos t\right)^{2}\right)+{\frac {\partial u}{\partial x}}\left(\left(e^{s}\cos t\right)+\left(-e^{s}\cos t\right)\right)+{\frac {\partial u}{\partial y}}\left(\left(e^{s}\sin t\right)+\left(-e^{s}\sin t\right)\right)\\&={\frac {\partial ^{2}u}{\partial x^{2}}}\left(e^{2s}\cos ^{2}t+e^{2s}\sin ^{2}t\right)+{\frac {\partial ^{2}u}{\partial x\partial y}}\left(2e^{2s}\cos t\sin t-2e^{2s}\sin t\cos t\right)+{\frac {\partial ^{2}u}{\partial y^{2}}}\left(e^{2s}\sin ^{2}t+e^{2s}\cos ^{2}t\right)+{\frac {\partial u}{\partial x}}\left(e^{s}\cos t-e^{s}\cos t\right)+{\frac {\partial u}{\partial y}}\left(e^{s}\sin t-e^{s}\sin t\right)\\&={\frac {\partial ^{2}u}{\partial x^{2}}}\left(e^{2s}\right)+0+{\frac {\partial ^{2}u}{\partial y^{2}}}\left(e^{2s}\right)+0+0\\&=\left(e^{2s}\right)\left({\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}\right)\\\end{aligned}}}
( e − 2 s ) ( ∂ 2 u ∂ s 2 + ∂ 2 u ∂ t 2 ) = ∂ 2 u ∂ x 2 + ∂ 2 u ∂ y 2 {\displaystyle \left(e^{-2s}\right)\left({\frac {\partial ^{2}u}{\partial s^{2}}}+{\frac {\partial ^{2}u}{\partial t^{2}}}\right)={\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}}
Q.E.D. (Ripples of 495 Years)
Okay, I was wrong. That was why we call it calc-u-lose D:
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