Hi, can someone help me out with the following question parts?(adsbygoogle = window.adsbygoogle || []).push({});

a) Let W be a compact region in R^3 bounded by a piecewise smooth closed surface S. Let [itex]f:W \to R[/itex] be a C^1 scalar function. Prove that for all constant vectors [itex]\mathop c\limits^ \to [/itex],

[tex]

\int\limits_{}^{} {\int\limits_S^{} f \mathop c\limits^ \to \bullet \mathop n\limits^ \to dS} = \int\limits_{}^{} {\int\limits_{}^{} {\int\limits_W^{} {\mathop c\limits^ \to \bullet \nabla fdV} } }

[/tex]

where [itex]\mathop n\limits^ \to [/itex] is the outward unit normal to S.

b) Let S be the surface of a tetrahedron in R^3, consisting of 4 triangular faces. Suppose that the ith face has area A_i and outward unit normal n_i. Show that

[tex]

\sum\limits_{i = 1}^4 {A_i \mathop {n_i }\limits^ \to } = \mathop 0\limits^ \to

[/tex]

[Hint: apply part (a) with a suitable choice of f.]

I can do part (a) but I can't figure out how to do part (b). In fact, I'm really unsure about what kind of a problem it is. For example, I know that part (a) is a divergence theorem question but the second one doesn't look familar.

Looking at what I need to show (the sum) I'm thinking that surface integrals might be involved. A surface integral is equal to the surface area over the appropriate region if the integrand is equal to one. I think I'm supposed to sum the 'vector' areas of the tetrahedron. I don't see how it can be zero unless the unit normals have opposite directions on opposing faces. But even that doesn't seem to lead anywhere because the area of each of the faces of the tetrahedron are not necessarily equal.

I don't know how to proceed. Can someone help me out? Thanks.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Vector Calculus

**Physics Forums | Science Articles, Homework Help, Discussion**