Given Two vector fields :
$$X(x,y,z)=(x,-y,-z)$$
$$Y(x,y,z)=(1,-y,x)$$
I want to calculate the dot product of these two vector fields $X.Y$. It's just that its vector fields that is confusing me a little but I assume its similar to calculating for two vectors. Here's what I did.
$X(x,y,z)=(-x,y,z)=xi-yj-zk$
$Y(x,y,z)=i-yj+xk$
$X.Y=(xi-yj-zk).(i-yj+xk)=x+y^2-zx$
This value is a scalar number at each pount (x,y,z).
Is this correct ?
If you say vector fields I would write $$X = x\frac{\partial }{\partial x}-y\frac{\partial }{\partial y}-z\frac{\partial}{\partial z}$$ and similarly for $Y$. The vector fields $\partial/\partial x$, etc. are differential operators acting on functions when you work on a generic manifold. The notation $i,j,k$ (or $e_1,e_2,e_3$) is used instead for three-dimensional vector spaces to denote the standard orthonormal basis. Anyway, if you are working in $\mathbb{R}^3$ and $i,j,k$ is the standard orthonormal basis, then your computation is correct.