div V

[akoaysigod]

V = r/|r| where r = xi + yj + zk
I know the answer is 2/r although I'm not sure how they get that.

d/dx(r/|r|) + d/dy(r/|r|) + d/dz(r/|r|) is supposed to equal 2/r

I tried putting in the values they gave for r but I just get a rather long expression and I can't make it look like 2/r; its likely I did something wrong or I'm not doing it correctly. I'm mostly confused as to how the vector r came back after the dot product.

 

[daon]

You have f(x,y,z) = Ui + Vj + Wk, where

U = x(x^2+y^2+z^2)^(-1/2)

V = y(x^2+y^2+z^2)^(-1/2)

W = z(x^2+y^2+z^2)^(-1/2)

dU/dx = (x^2+y^2+z^2)^(-1/2) - x^2*(x^2+y^2+z^2)^(-3/2)

(y^2+z^2) / (x^2+y^2+z^2)^(3/2)

Similarly for the others.

When we add these, we have:

\(\displaystyle \frac{y^2+z^2}{(x^2+y^2+z^2)^{3/2}}+\frac{x^2+z^2}{(x^2+y^2+z^2)^{3/2}}+\frac{x^2+y^2}{(x^2+y^2+z^2)^{3/2}} = \frac{2(x^2+y^2+z^2)}{(x^2+y^2+z^2)^{3/2}}\)