How to Evaluate Line, Surface, and Volume Integrals with Vector Functions

[shermaine80]

May i know the general procedures for evaluating the following line,surface and volume for the following:

(1) triple integrate vector A.dV
(2) Double integrate vector A.n.dS
(3) integrate vector A.dr

 

[HallsofIvy]

Whole books are written on this!

Also, I don't know what you mean by "A.dV" dV is a scalar quantity, not a vector so you cannot take the dot product of a vector, A, with it.

If you are given a surface, S, you can always write it in terms of parametric equations, in terms of two parameters, say u and v: x(u,v), y(u,v), z(u,v). You can then write it as a vector equation in an obvious way: $\vec{r}(u,v)= x(u,v)\vec{i}+ y(u,v)\vec{j}+ z(u,v)\vec{k}$. The "fundamental vector product" is the cross product of the two partial derivatives: $\vec{r}_u\times\vec{r}_v$ and the "vector differential of surface area" is $\vec{r}_u\times\vec{r}_v dudv$. Of course, that points in opposite directions depending on the order of multiplication: that's because you need to determine an orientation of the surface.

For a path, which depends on one parameter, say t, $\vec{r}= x(t)\vec{i}+ y(t)\vec{j}+ z(t)\vec{k}$, we have $d\vec{r}= x'\vec{i}dx+ y'\vec{j}dy+ z'\vec{k}dz$. A.dr is the dot product of A with that.