- #1

N00813

- 32

- 0

## Homework Statement

Using the formal limit definition of the derivative, derive expressions for the Fourier Transforms with respect to x of the partial derivatives [itex] \frac{\partial u}{\partial t} [/itex] and [itex] \frac {\partial u}{\partial x} [/itex].

## Homework Equations

The Fourier Transform of a function [itex] u(x,t) [/itex] is:

[tex]

\tilde{u}(k,t) = \int_{-\infty}^{\infty} u(x,t) e^{-ikx} dx

[/tex]

## The Attempt at a Solution

I attempted to write the FT of the derivative with respect to time as [itex] \lim_{dt \to 0} \frac{u(x,t+dt) - u(x,t)}{dt} [/itex] and then Fourier Transform it. Pulling the limit outside the integral, I got [itex] FT of (\frac{\partial u}{\partial t}) = \tilde{\frac {\partial u}{\partial t}}(k,t). [/itex]Now, with the partial derivative of x, I'm not so sure. The partial derivative wrt x is [itex] \lim_{dx \to 0} \frac{u(x + dx,t) - u(x,t)}{dx} [/itex]. When placed in the Fourier transform, I don't know what I can do. I don't think the dx at the end of the integral cancels out the dx in the definition of the derivative.