# Integral - what does it represent?

by Benny
Tags: integral, represent
 P: 585 Hi, I am given the following integral. $$\int\limits_{}^{} {\int\limits_S^{} {\mathop F\limits^ \to } } \bullet d\mathop S\limits^ \to = \int\limits_{}^{} {\int\limits_D^{} {\mathop F\limits^ \to \bullet \mathop n\limits^ \to } } dS$$ The n vector is an outward unit normal. So does the RHS of the above represent how much 'stuff' is coming out of a surface? Or does it have something to do with what is happening on a surface? The book says that the integral is called the flux of F across S. In many examples, there are little arrows point out of the surface so I'm not sure what the integral is supposed to represent. Any help would be good thanks.
 HW Helper P: 1,025 The form you put above is still a surface integral on either side, further work is required to change such into a double integral over a domain D in the xy-plane; one such way is to parameterize the surface by, say $\vec{r}(u,v)$ where the parameters u and v range over values in D, the formula is then $$\int\limits_{}^{} {\int\limits_S^{} {\mathop F\limits^ \to } } \bullet d\mathop S\limits^ \to = \int\limits_{}^{} {\int\limits_D^{} {\mathop F\limits^ \to \left( \vec{r}(u,v) \right) \bullet \left( \vec{r}_{u} \times \vec{r}_{v}\right) } } dA$$
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,340 $$\int\limits_{}^{} {\int\limits_S^{} {\mathop F\limits^ \to } } \bullet d\mathop S\limits^ \to = \int\limits_{}^{} {\int\limits_S^{} {\mathop F\limits^ \to \bullet \mathop n\limits^ \to } } dS$$ Just different ways of writing the same thing: $$d\mathop S\limits^ \to$$ is defined as $$\mathop n\limits^ \to } } dS$$ Yes, it can be interpreted as "flux of F across S"- that is, the rate at which some property (whatever F represents) is flowing through surface S. One way to interpret the Divergence theorem: $$\int\int\int\limits_B^{ } (\nabla \bullet \mathop v\limits^ \to )dV= \int\int_S \mathops v\limit^ \to \bullet d\mathop \sigma\limit^ \to$$ is that the amount of something inside region B depends upon how much is flowing in or out through the surface S.