User:Kamek77

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[edit] Partial Derivativasdfadsfa

For u(x,y) where x = x(s,t), y = y(s,t):


\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}


\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:

[edit] Phantasm Stage: 15.5.48

Let u = f(x,y), x = escost, y = essint.
Show that 
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}
.


\frac{\partial x}{\partial s} = e^s \cos t
\frac{\partial x}{\partial t} = -e^s \sin t

\frac{\partial^2 x}{\partial s^2} = e^s \cos t
\frac{\partial^2 x}{\partial t^2} = -e^s \cos t

\frac{\partial y}{\partial s} = e^s \sin t
\frac{\partial y}{\partial t} = e^s \cos t

\frac{\partial^2 y}{\partial s^2} = e^s \sin t
\frac{\partial^2 y}{\partial t^2} = -e^s \sin t

\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}
\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}

\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}
\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}

\begin{align}
\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{align}
\begin{align}
\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{align}

\begin{align}
\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{align}
\begin{align}
\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{align}

\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}
\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}

\begin{align}
\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{align}

\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.


Okay, I was wrong. That was why we call it calc-u-lose D: