# Homework Help: Deriving expressions for Fourier Transforms of Partial Derivatives

1. Mar 24, 2014

### N00813

1. The problem statement, all variables and given/known data

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

2. Relevant equations

The Fourier Transform of a function $u(x,t)$ is:
$$\tilde{u}(k,t) = \int_{-\infty}^{\infty} u(x,t) e^{-ikx} dx$$

3. The attempt at a solution

I attempted to write the FT of the derivative with respect to time as $\lim_{dt \to 0} \frac{u(x,t+dt) - u(x,t)}{dt}$ and then Fourier Transform it. Pulling the limit outside the integral, I got $FT of (\frac{\partial u}{\partial t}) = \tilde{\frac {\partial u}{\partial t}}(k,t).$

Now, with the partial derivative of x, I'm not so sure. The partial derivative wrt x is $\lim_{dx \to 0} \frac{u(x + dx,t) - u(x,t)}{dx}$. 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.

2. Mar 24, 2014

### pasmith

You should avoid using "dx" and so forth as anything other than part of a derivative or integral (or as a differential form). Otherwise you run into this sort of confusion. When taking limits, use either $\delta x$ or another letter.

You want to calculate
$$\int_{-\infty}^\infty \lim_{h \to 0} \frac{u(x+h, t) - u(x,t)}h e^{-ikx}\,dx = \lim_{h \to 0} \frac1h \left( \int_{-\infty}^\infty u(x + h, t)e^{-ikx}\,dx - \int_{-\infty}^\infty u(x, t)e^{-ikx}\,dx \right)$$

3. Mar 24, 2014

### N00813

Thanks, I must have derped out last night.

I'm thinking about turning the 2nd term into $\lim_{h \to \infty} \frac{1}{h} \tilde{u}(k,t)$ and substituting the y = x + h into the first term, to give a final result of:
$$\lim_{h \to 0} \frac{1}{h} ((e^{ik(0+h)}-e^{ik0}) \tilde{u}) = ike^{0} \tilde{u}$$

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