1. ## Vector Integral?

Hey guys!

I'm working through a proof from an old paper for my project and I'm really stumped with this notation. I want to understand the meaning of:

For $x$, $y \in S^{2}$, $k \ne l$
$\int_{S^2} P_k({x}{z})P_l({y}{z}) dz=0$
Where the $P_k$ are Legendre polynomials degree $k$.

Now I know that this is a kind of more generalised orthogonality relation and the legendre polynomials take scalar arguments (at least I think I know that...) in which case I think it would mean

For $\boldsymbol{x}$, $\boldsymbol{y} \in S^{2}$, $k \ne l$
$\int_{S^2} P_k(\boldsymbol{x}\cdot \boldsymbol{z})P_l(\boldsymbol{y} \cdot \boldsymbol{z}) d\boldsymbol{z}=0$
But then I dont understand the meaning of $d\boldsymbol{z}$.

Ive read in an old text that used the notation "for $x=(x_1,x_2, \ldots, x_n) ...$where $dx=dx_1 dx_2 \cdots dx_n$ is the ordinary Lebesque measure", but I haven't been taught the Lebesque stuff and it looks real tricky...

So if anyone could shed a little light on whats going on, or point me to somewhere that does if its elementary (and im just being stupid) then id be so grateful!

2. ## Re: Vector Integral?

The "z" is the variable of integration and so is a "dummy" variable. It means "for constant vectors x and y, take the dot product with the vector z (so that the arguments of the Legendre functions are numbers), then integrate over the plane".

3. ## Re: Vector Integral?

When you say integrate over the plane, does that make sense when $S^2$ is the surface of the sphere so we have a surface integral?

4. ## Re: Vector Integral?

Solved Turns out the guy that wrote the paper was just using really wierd notation for something sensible. For some reason he was using $dz$ instead of $d^2 \Omega$ the usual area element on a sphere so it is just a lovely scalar function integrated over the surface of a sphere .
Thanks anyway!

5. ## Re: Vector Integral?

by the way so when you take the surface integral of the dot product of a vector function dotted into a normal vector (or other function)-for the dot product you get the component of the vector function in the normal direction, but to find the surface area of a surface don't you only need the area of the face? eg electromagnetism there is a dot product of vector function F(x,y,z) dot n
also why is it important that both sides of the face in the limit go to zero?
can you define surface integral as the limit sum like, it seems that you only need the area of the face without dot product
$\sum_{I=1}^{\infty }F_{i}(x,y,z)\cdot \hat{n}\Delta S_{i}$ I guess sometimes you want to project different parts of a surface onto different regions in the coordinate planes, xy, yz, etc...? Thanks very much!