Solution: ▽ · F = −3xy / (x2 + y2)^5/2

[scholesmu]

Find ▽ · F if F = x i / (x2 + y2)^3/2 + y j (x2 + y2)^3/2 .

Solution: ▽ · F = −1 / (x2 + y2)^3/2

 

[waht]

Find the partial derivatives of F with respect to x, y and z and then add them up.

 

[HallsofIvy]

Find ▽ · F if F = x i / (x2 + y2)^3/2 + y j (x2 + y2)^3/2 .

Solution: ▽ · F = −1 / (x2 + y2)^3/2

The derivative of x/(x^2+ y^2)^{3/2}= x(x^2+ y^2)^{-3/2} with respect to x, by the product rule, is (x^2+ y^2)^{-3/2}- (3/2)(x)(2x)(x^2+ y^2)^{-5/2}
= (x^2+ y^2)(x^2+ y^2)^{-5/2}- 3x^2(x^2+ y^2)^{-5/2}= (y^2- 2x^2)/(x^2+ y^2)^{5/2}.

Now, what is the derivative of y/(x^2+ y^2)^{3/2}?