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Year 1 of the natural sciences course of the University of Cambridge

Mathematics course of the year 1 of the natural sciences course of the University of Cambridge

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge

2023

Paper 1

The <chain rule> expresses the coordinate <differential operators> as
$$
\partial_x=a\partial_\xi+b\partial_\eta,\qquad \partial_y=-b\partial_\xi+a\partial_\eta.
$$
The coefficients are constant, so a second application gives
$$
u_{xx}=a^2u_{\xi\xi}+ab(u_{\xi\eta}+u_{\eta\xi})+b^2u_{\eta\eta},
$$
and
$$
u_{yy}=b^2u_{\xi\xi}-ab(u_{\xi\eta}+u_{\eta\xi})+a^2u_{\eta\eta}.
$$
Adding cancels the mixed <partial derivatives>, and $a^2+b^2=1$ gives
$$
\boxed{u_{xx}+u_{yy}=u_{\xi\xi}+u_{\eta\eta}.}
$$
The transformation has <rotation matrix> $\begin{pmatrix}a&-b\\b&a\end{pmatrix}$, so this identity expresses the <rotational invariance of the Laplacian>.

Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 14 / d / Solution

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 14 / d / Solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 14 / d / Solution

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