 Vector areas of a closed polyhedron sum to zero

ID: vector-areas-of-a-closed-polyhedron-sum-to-zero

For the faces of a closed polyhedron, take outward unit normal vectors. Applying the divergence theorem to an arbitrary constant vector field

c

gives

c \cdot \sum_{f} A_{f} \hat{n}_{f} = \int_{\partial Ω} c \cdot \hat{n} d A = \int_{Ω} \nabla \cdot c d V = 0.

Since

c

is arbitrary, the vector areas sum to zero. This is an oriented statement; the positive scalar surface areas do not cancel. The same proof works for every piecewise smooth closed boundary to which the divergence theorem applies.

 New to topics? Read the docs here!