Tensor-product inheritance of geometric basis properties (${\partial }_u^p{\partial }_v^q({\varphi }_i(u){\psi }_j(v))={\varphi }_i^{(p)}(u){\psi }_j^{(q)}(v)$)

Nonnegative tensor-product surface basis factors yield nonnegative product weights. Their two partition of unity sums multiply to one. If the factors are $C^r$ and $C^s$, mixed derivatives through separate orders $r,s$ are continuous, so the resulting finite or locally finite surface representation is jointly $C^{\min (r,s)}$. Control coefficients can cancel derivative jumps and yield more smoothness; inherited continuity is a guaranteed lower bound, not a claim that every surface has exactly the basis's worst regularity.