Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Sign up to join this community
Can somebody give me hints to solve the following question?
I need to find the closed surface integral (using divergence theorem) of
$$\oint \vec{r} (\vec{a} \cdot \vec{n}) da$$
where $\vec{n}$ is the outward pointing unit vector of the surface S aroud the volume V and $\vec{r}$ is the radius vector and $\vec{a}$ is a constant vector
Thanks.
I am a bit unsure, if I treat the radius vector correctly, and I assume $da$ is a surface element of $S$ and has nothing to do with the constant vector $a$, so it might be like this: $$ \int r (a \cdot n) da = \\ \int (x e_x + y e_y + z e_z) (a \cdot n) da = \\ e_x \int x (a \cdot n) da + e_y \int y (a \cdot n) da + e_z \int z (a \cdot n) da = \\ e_x \int ((xa) \cdot n) da + e_y \int ((ya) \cdot n) da + e_z \int ((za) \cdot n) da = \\ e_x \int (\mbox{div } x a) dV + e_y \int (\mbox{div } y a) dV + e_z \int (\mbox{div } z a) dV = \\ e_x \int a_x dV + e_y \int a_y dV + e_z \int a_z dV = \\ \int (e_x a_x + e_y a_y + e_z a_z) dV = \\ \int a dV = \\ V a $$
