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Products of two univariate bases form a surface basis. Nonnegativity, partition of unity and coordinate-wise linear precision pass to the products by multiplication of the respective univariate identities. If the factors are $C^r$ and $C^s$, each mixed derivative through orders $r,s$ is the product of the corresponding derivatives, giving continuous mixed derivatives and joint $C^{\min (r,s)}$ regularity.

For nonnegative barycentric parameters, these total-degree Bernstein basis functions define a surface over a triangle. The triangular control net has $(n+1)(n+2)/2$ points. Its index constraint couples the three exponents, so it is not a tensor product of two independent univariate bases. The multinomial theorem gives nonnegativity and partition of unity.