Question 2

One sufficient version of the divergence theorem is the following. Let

R

be a bounded region in

R^{3}

with a closed, piecewise smooth surface as its boundary, and let

F

be a continuously differentiable function with values in

R^{3}

on an open neighbourhood of the closure of

R

. With

n

the outward unit normal vector,

∭_{R} \nabla \cdot F d V = \iint_{\partial R} F \cdot n d S .

(1)

The boundary surface integral is the total outward flux integral. If the boundary has several components, each has the outward orientation relative to the material of

R

; inner components therefore point into the holes. Edges and isolated vertices may be omitted when assigning the unit normal vector, since they have zero surface area.

Solution

 0  0

For each of the three constant Cartesian unit vectors

e_{j}

, apply the divergence theorem to the vector field

ϕ e_{j}

. Its divergence is

\partial_{j} ϕ

, so

∭_{R} \partial_{j} ϕ d V = \iint_{\partial R} ϕ n_{j} d S (j = 1, 2, 3) .

(1)

These are precisely the three components of

∭_{R} \nabla ϕ d V = \iint_{\partial R} ϕ n d S = \iint_{\partial R} ϕ d S .

(2)

Thus the volume integral of the gradient is the boundary surface integral weighted by the vector surface element

d S = n d S

. The regularity needed here is that

ϕ

be continuously differentiable on a neighbourhood of the closure of

R

Adding the zero base contribution agrees with the volume integral, verifying the divergence theorem. The apex has zero surface area; the vector field is smooth there, so no additional flux is hidden at that point.