To stimul should be able to obtain solutions to certain important PDEs using a variety of techniques e.g. Green’s functions, separation of variables, and in cases interpret these in physical terms. They should also be familiar with important analytic properties of the solution.

这是一份Bath巴斯大学MA40059作业代写的成功案

$$
\begin{gathered}
Y(x, \epsilon)=y(x)=\epsilon \eta(x), \
Y^{\prime}(x, \epsilon)=y^{\prime}(x)+\epsilon \eta^{\prime}(x) .
\end{gathered}
$$
Then the meaning of $\delta y$ is
$$
\delta y=\left(\frac{\partial Y}{\partial \epsilon}\right){\epsilon=0} d \epsilon=\eta(x) d \epsilon ; $$ this is just like a differential $d Y$ if $\epsilon$ is the variable. The meaning of $\delta y^{\prime}$ is $$ \delta y^{\prime}=\left(\frac{\partial Y^{\prime}}{\partial \epsilon}\right){\epsilon=0} d \epsilon=\eta^{\prime}(x) d \epsilon .
$$
This is identical with
$$
\frac{d}{d x}(\delta y)=\frac{d}{d x}[\eta(x) d \epsilon]=\eta^{\prime}(x) d \epsilon
$$

MA40059 COURSE NOTES ：

We can show that these nine quantities are the components of a second-rank tensor which we shall denote by UV. Note that this is not a dot product or a cross product; it is called the direct product of $\mathrm{U}$ and $\mathrm{V}$ (or outer product or tensor product). Since $\mathbf{U}$ and $\mathbf{V}$ are vectors, their components in a rotated coordinate system are:
$$
U_{k}^{\prime}=\sum_{i=1}^{3} a_{k i} U_{i}, \quad V_{l}^{\prime}=\sum_{j=1}^{3} a_{l j} V_{j} .
$$
Hence the components of the second-rank tensor UV are
$$
U_{k}^{\prime} V_{l}^{\prime}=\sum_{i=1}^{3} a_{k i} U_{i} \sum_{j=1}^{3} a_{l j} V_{j}=\sum_{i, j=1}^{3} a_{k i} a_{l j} U_{i} V_{j}
$$
which is just with $T_{i j}=U_{i} V_{j}$ and $T_{k l}^{\prime}=U_{k}^{\prime} V_{l}^{\prime}$.