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I am starting my studies in antenna theory and need help to prove the following identities:

I) $\nabla^{'}\left[\mathbf{J}(\mathbf{r^{'}})e^{j\mathbf{k}\cdot\mathbf{r^{'}}}\right] = j\mathbf{k}\cdot\mathbf{J}(\mathbf{r^{'}})e^{j\mathbf{k}\cdot\mathbf{r^{'}}} - j\omega\rho(\mathbf{r^{'}})e^{j\mathbf{k}\cdot\mathbf{r^{'}}}$ and

II) $\mathbf{k}\cdot\int_V\mathbf{J}(\mathbf{r^{'}})e^{j\mathbf{k}\cdot\mathbf{r^{'}}} d^3\mathbf{r^{'}} = \omega \int_V\rho(\mathbf{r^{'}})e^{j\mathbf{k}\cdot\mathbf{r^{'}}}d^3\mathbf{r^{'}}$

Notation and hypotheses:

1) $\mathbf{J}$ and $\rho$ are the current and charge densities, respectively, and both are restricted to the integral volume V (i.e. outside V both are null);

2) The notation for the distance $\mathbf{r^{'}}$ is as follows:

Absolute and relative distances notation

3) $\mathbf{k} = k\mathbf{\hat{k}}$, with $k=\omega \sqrt{\mu \epsilon} = \omega/c$, c is light velocity, $\mu$ and $\epsilon$ electromagnetics constants, $\omega$ wave frequency $\mathbf{\hat{k}}$ is the unit vector in the direction of the wave propagation.

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  • $\begingroup$ The second one is the continuity equatio in fourier space. You need to be more explicit about time dependency. Probably rho is something like rho hat time exp iwt. The time dependecy of rho and J is the key to this question. Look it up in the book how tjey are defining it. $\endgroup$ – lalala Jun 1 '19 at 19:56
  • $\begingroup$ What are your thoughts on the problem, you should show some attempt at a solution so we can understand exactly what is tripping you up $\endgroup$ – Triatticus Jun 1 '19 at 20:13

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