User:Kamek77

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:

Phantasm Stage: 15.5.48
''Let $$u = f(x,y)$$, $$x = e^s \cos t$$, $$y = e^s \sin t$$.  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. (Ripples of 495 Years)

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