I've just started my Galaxy Dynamics module and we are refreshing ourselves on Divergence and Curl, etc etc. I've come across a divergence example which I can't quite understand how to handle:
So I have a force given by
$$ F = (x^2 + y^2 + z^2)^n(xi+yj+zk)$$
I was wondering how we handle this for divergence...an explanation of the setup would be excellent.
I'm not sure how to handle the $(xi+yj+zk)$ with respect to $(x^2 + y^2 + z^2)^n$...?
On
The product rule essentially gives $$ \operatorname{div}(f \mathbf{v}) = (\operatorname{grad} f) \cdot \mathbf{v} + f \operatorname{div} \mathbf{v}, $$ so we have $$ \operatorname{div}( (x^2+y^2+z^2)^n (x \mathbf{e}_1 + y \mathbf{e}_2 + z \mathbf{e}_3 ) ) = (x \mathbf{e}_1 + y \mathbf{e}_2 + z \mathbf{e}_3 ) \cdot \nabla(x^2+y^2+z^2)^n + (x^2+y^2+z^2)^n ( \frac{\partial}{\partial x} x + \frac{\partial}{\partial y} y + \frac{\partial}{\partial z} z ) $$ The latter is obviously just $3(x^2+y^2+z^2)^n$. The former derivative is $$ n(x^2+y^2+z^2)^{n-1} \nabla(x^2+y^2+z^2), $$ and the last derivative here is $$ 2x \mathbf{e}_1+2y \mathbf{e}_2 + 2z \mathbf{e}_3; $$ dotting gives $$ 2(x^2+y^2+z^2)n(x^2+y^2+z^2)^{n-1}, $$ and you can carry on from here.
Hint:
the divergence of a vector filed $\mathbf{F}=A\mathbf{i}+B\mathbf{j}+C\mathbf{k}$ is defines as: $$ div (\mathbf{F})=\dfrac{\partial A}{\partial x}+\dfrac{\partial B}{\partial y}+\dfrac{\partial C}{\partial z} $$ in your case: $$ A=x(x^2+y^2+z^2)^n \quad B=y(x^2+y^2+z^2)^n \quad C=z(x^2+y^2+z^2)^n $$