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# Tensor Products of Integration Rules

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Integration rules for rectangles

can be obtained by forming tensor products of univariate formulas.

If the formulas for approximating are exact for polynomials of degree , then the product rule

is exact for polynomials of coordinate degree .

For simplicity, we consider functions of two variables (). To this end we denote by

the two univariate integration rules. Then, the bivariate formula has the form

If is a polynomial with coordinate degree ,

then, for fixed , the function is a polynomial of degree in . Hence, the expression in brackets coincides with the integral . Interchanging summation and integration, we obtain

Again, the sum equals the corresponding integral, and it follows that

which is the anticipated value for .

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 automatisch erstellt am 17.  1. 2017