Let Ω be a body and let be its surface. Let be the normal to the surface. Let be a vector field on Ω and let be a secondorder tensor field on Ω. Show that
Proof:
Recall the relation
Integrating over the volume, we have
Since and are constant, we have
From the divergence theorem,
we get
Using the relation
we get
Since and are constant, we have
Therefore,
Since and are arbitrary, we have
Let Ω be a body and let be its surface. Let be the normal to the surface. Let be a vector field on Ω. Show that
Proof:
Recall that
where is any secondorder tensor field on Ω. Let us assume that . Then we have
Now,
where is any secondorder tensor. Therefore,
Rearranging,
