Final answer:
To show that the given integrand is exact in the domain D, we need to check if its differential form is exact. The given differential form can be rewritten in terms of the total differential of a function and we can use Clairaut's theorem to check if it is exact.
Step-by-step explanation:
To show that the given integrand is exact, we need to check if its differential form is exact in the domain D. The given differential form is (ysinzdx+xsinzdy+xycoszdz).
We can rewrite this in terms of the total differential of a function F(x,y,z) as dF = Pdx + Qdy + Rdz, where P = ysinz, Q = xsinz, and R = xycosz.
To check if the differential form is exact, we need to ensure that its partial derivatives satisfy the conditions of Clairaut's theorem, which states that the mixed partial derivatives must be equal.
In this case, we have:
∂P/∂y = sinz, ∂Q/∂x = sinz, ∂Q/∂y = 0, ∂R/∂x = ycosz, ∂R/∂z = -xysinz.
Since all the mixed partial derivatives are equal, the differential form is exact in the domain D.