Question: dotproduct 's pde equations return nothing and coefficient question

Calcution-A

with(VectorCalculus):
with(LinearAlgebra):
f := [x3*(1+x2),x1,x2*(1+x1)]:
g := [0,1+x2,-x3]:
fv := <x3*(1+x2),x1,x2*(1+x1)>:
gv := <0,1+x2,-x3>:
ad_f_g :=  MatrixAdd(MatrixVectorMultiply(Jacobian(g,[x1,x2,x3]),fv),-MatrixVectorMultiply(Jacobian(f,[x1,x2,x3]),gv)):
b := <Diff(w(x1,x2,x3),x1),Diff(w(x1,x2,x3),x2),Diff(w(x1,x2,x3),x3)>:
gv := <0,1+x2,-x3>:
c1 := DotProduct(b, gv):
c2 := DotProduct(b, ad_f_g):
with(PDEtools, casesplit, declare):
declare(w(x1,x2,x3)):
ex1 := {c1,c2};
pdsolve(ex1)

Calcution-B

with(PDEtools, casesplit, declare):
ex1 := {Diff(w(x1,x2,x3),x2)*(1+x2) - Diff(w(x1,x2,x3),x3)*x3 = 0,
Diff(w(x1,x2,x3),x2)*x1+Diff(w(x1,x2,x3),x3)*(1+x1)*(1+2*x2)=0}:
pdsolve(ex1)

only explicit write equations in ex1, it solve _F1(x1)

answer should be w = x1

1. using Calculation-A, dotproduct return equation having a bar above some terms
   and pdsolve return nothing, is this bar influence it,
   if explicit write equations like Calculation-B, it can solve, How to fix the problem in Calculation-A

2. why psolve not return x1, i know _F1 is cofficient, however, there is no initial condition
   how can i know _F1 is 1 in maple 15