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. Please be reminded that a test function is needed as a part of the integr and in order to prove any delta function identities. Prove the following: \text { Prove that } \delta(a x)=\frac{1}{|a|} \delta(x)_{2} a \neq 0 Use the iedntity in part (a) to prove that \delta[g(x)]=\sum_{m} \frac{1}{\left|g^{\prime}\left(x_{m}\right)\right|} \delta\left(x-x_{m}\right) \text { where } g\left(x_{m}\right)=0 \text { and } g^{\prime}\left(x_{m}\right) \neq 0 \text {. } ) Show that 6(r)/r = −8′(r) when it appears as part of the integr and of a three-dimensional integral in spherical coordinates. Convince yourself that the test function f(r) does not provide any information. Then try f(r)/r. \text { (d) Show that } \nabla \cdot[\delta(r-a) \hat{r}]=\left(a^{2} / r^{2}\right) \delta^{\prime}(r-a) \text { when it appears as part of the integrand of a three- } dimensional integral in spherical coordinates.

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