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This is more of a general question, but I've encountered this kind of exercises a lot in my current preperations for my exam:

There are two cases but the excercise is pretty much the same:

Compute

$$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace \vec{A}(\vec{r})=(\vec{a} \cdot \vec{r}) \vec{r} \qquad \vec{a} = const.$$

$$(2) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace \vec{A}(\vec{r})=(\vec{a} \times \vec{r})\times \vec{r} \qquad \vec{a} = const.$$

My attempt to these was to always write down the expression that would come out. I will just elaborate this for (1) but I think one gets the gist of it:

$$\nabla \cdot \vec{A}(\vec{r})= \frac{\partial}{\partial x}(a_xr_x^2+a_yr_yr_x+a_zr_zr_x)+\frac{\partial}{\partial y}(...)+\frac{\partial}{\partial z}(...)$$

As one can see this is very tedious and exhausting to do, especially during the exam, also the results are very vagues, since it depends on how the derivative of r looks like. So I am wondering, if there is a more elegant way to solve these kind of exercises.

My assumption is, that there are some neat little tricks concering the problem of having a dotproduct/crossproduct of the same vector and a constant vector. Sadly I couldn't really find anything in my lecture notes, nor on the internet.

Thanks in advance :)

P.S.: I hope this is the right place to post this question since I wasn't sure if this is rather a vector analysis or a calculus question

There are two cases but the excercise is pretty much the same:

Compute

$$(1) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace \vec{A}(\vec{r})=(\vec{a} \cdot \vec{r}) \vec{r} \qquad \vec{a} = const.$$

$$(2) \space \operatorname{div}\vec{A}(\vec{r}) \qquad , where \thinspace \vec{A}(\vec{r})=(\vec{a} \times \vec{r})\times \vec{r} \qquad \vec{a} = const.$$

My attempt to these was to always write down the expression that would come out. I will just elaborate this for (1) but I think one gets the gist of it:

$$\nabla \cdot \vec{A}(\vec{r})= \frac{\partial}{\partial x}(a_xr_x^2+a_yr_yr_x+a_zr_zr_x)+\frac{\partial}{\partial y}(...)+\frac{\partial}{\partial z}(...)$$

As one can see this is very tedious and exhausting to do, especially during the exam, also the results are very vagues, since it depends on how the derivative of r looks like. So I am wondering, if there is a more elegant way to solve these kind of exercises.

My assumption is, that there are some neat little tricks concering the problem of having a dotproduct/crossproduct of the same vector and a constant vector. Sadly I couldn't really find anything in my lecture notes, nor on the internet.

Thanks in advance :)

P.S.: I hope this is the right place to post this question since I wasn't sure if this is rather a vector analysis or a calculus question

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