 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/5/ii/solution

Under regularity and integrability conditions permitting the limits to pass through the integral, apply the difference quotient directly in time:

F_{x} [u_{t}] (k, t) = h \to 0 lim \int_{R} \frac{u ( x , t + h ) - u ( x , t )}{h} e^{- ik x} d x = h \to 0 lim \frac{u ( k , t + h ) - u ( k , t )}{h} = \partial_{t} u (k, t) .

For the spatial derivative, translate the integration variable in the first term:

F_{x} [u_{x}] (k, t) = h \to 0 lim \int_{R} \frac{u ( x + h , t ) - u ( x , t )}{h} e^{- ik x} d x = h \to 0 lim \frac{e ^{ikh} - 1}{h} u (k, t) = ik u (k, t) .

Consequently

F_{x} [u_{t}] = \partial_{t} u, F_{x} [u_{x}] = ik u .

The second calculation is the Translation property of the Fourier transform applied to a difference quotient. These are the Fourier transform of a derivative identities. The stated decay removes boundary terms in the alternative integration-by-parts derivation; rigorous interchange of limits also requires suitable control, for example integrable derivatives. The formulas extend to tempered distributions.

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