Cauchy Riemann Equations

[wolly]

I have to find [math]\frac{\partial t}{\partial \varphi}[/math][math]t=x^2+y^2[/math]and [math]\varphi(t(x,y))=x^2+y^2[/math]How did 4x^2 appear?

 

[wolly]

Also how did that [math]e^{-t}[/math] appear from?

 

[logistic_guy]

View attachment 39359
I have to find [math]\frac{\partial t}{\partial \varphi}[/math][math]t=x^2+y^2[/math]and [math]\varphi(t(x,y))=x^2+y^2[/math]How did 4x^2 appear?

You have two functions:
\(\displaystyle \varphi(t)\)
\(\displaystyle t(x,y)\)

You want to find \(\displaystyle \frac{\partial^2 \varphi}{\partial x^2}\)

Apply the chain rule.

\(\displaystyle \frac{\partial \varphi}{\partial x} = \frac{\partial \varphi}{\partial t}\frac{\partial t}{\partial x} = 2x\frac{\partial \varphi}{\partial t}\)

You can change \(\displaystyle \frac{\partial \varphi}{\partial t}\) to \(\displaystyle \frac{d \varphi}{d t}\) if you want because it has only one variable.

\(\displaystyle \frac{\partial^2 \varphi}{\partial x^2} = \frac{\partial}{\partial x}\left(2x\frac{\partial \varphi}{\partial t}\right)\)

Apply the product rule.

\(\displaystyle \frac{\partial^2 \varphi}{\partial x^2} = 2\frac{\partial \varphi}{\partial t} + 2x\frac{\partial}{\partial x}\left(\frac{\partial \varphi}{\partial t}\right)\)

Apply the chain rule.

\(\displaystyle \frac{\partial^2 \varphi}{\partial x^2} = 2\frac{\partial \varphi}{\partial t} + 2x\frac{\partial}{\partial t}\left(\frac{\partial \varphi}{\partial t}\right)\frac{\partial t}{\partial x} = 2\frac{\partial \varphi}{\partial t} + 2x\frac{\partial^2 \varphi}{\partial t^2}2x = 2\frac{\partial \varphi}{\partial t} + 4x^2\frac{\partial^2 \varphi}{\partial t^2}\)

That's how \(\displaystyle 4x^2\) appeared.

 

[logistic_guy]

View attachment 39360
Also how did that [math]e^{-t}[/math] appear from?

What was the question that made you did these calculations? Can you write it?