Extension of Taylors theorem

Taylor s theorem may be extended so as to include the expansion of functions of two or more independent variables. Let 

If y now becomes y + k, each term of equation (2) will change so that 

The final result is exactly the same whether we expand first with respect to y or in the reverse order. 

By equating the coefficients of hk in the identical results ob- tained by first expanding with regard to h, (2) above, and by first expanding with regard to k, we get 

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