Vector Calculus | Part IA, 2001

State the divergence theorem for a vector field u(r)\mathbf{u}(\mathbf{r}) in a closed region VV bounded by a smooth surface SS.

Let Ω(r)\Omega(\mathbf{r}) be a scalar field. By choosing u=cΩ\mathbf{u}=\mathbf{c} \Omega for arbitrary constant vector c\mathbf{c}, show that

VΩdv=SΩdS\int_{V} \nabla \Omega d v=\int_{S} \Omega d \mathbf{S}

Let VV be the bounded region enclosed by the surface SS which consists of the cone (x,y,z)=(rcosθ,rsinθ,r/3)(x, y, z)=(r \cos \theta, r \sin \theta, r / \sqrt{3}) with 0r30 \leq r \leq \sqrt{3} and the plane z=1z=1, where r,θ,zr, \theta, z are cylindrical polar coordinates. Verify that ()(*) holds for the scalar field Ω=(az)\Omega=(a-z) where aa is a constant.

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