  Starting with a regular symmetric Dirichlet form on a locally compact
separable metric space $X$, our paper studies elements of vector analysis,
$L_p$-spaces of vector fields and related Sobolev spaces. These tools are then
employed to obtain existence and uniqueness results for some quasilinear
elliptic PDE and SPDE in variational form on $X$ by standard methods. For many
of our results locality is not assumed, but most interesting applications
involve local regular Dirichlet forms on fractal spaces such as nested fractals
and Sierpinski carpets.
