So I kind of get both methods of proof: http://math.gmu.edu/~memelian/teaching/Fall11/math678/hw/hw1sol.pdf
But I'm having issues reconciling the definition of the Laplace operator as the sum of second partial derivatives with the vector form definition (Dx(v)*Dx(v)) which seems like it would be the sum of squared first partial derivatives... anyone care to set me straight? Am I forgetting something basic?
On
You're right that $Du \cdot Du$ is the sum of squares of first derivatives and not the Laplacian. I'm guessing this is a typo and it's meant to be $D_x \cdot D_x u$ - if you remove the first $u$ from each expression in the chain of equalities then what you get looks like the "vector form" of the coordinate argument, provided you permit a little abuse of notation.
Where did the "vector form definition" come from? Sometimes you see a term of this form (called the Dirichlet energy) integrated over a closed, compact manifold $M$ to get, by integration by parts, $$\int_M \nabla v \cdot \nabla v = \int_{\partial M} (\nabla v \cdot \hat{n})\nabla v - \int_M v \Delta v = -\int_M v\Delta v$$ but notice that this equation is only true integrated over $M$, not pointwise.