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 Exercise [10.04] 
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Joined: 12 Mar 2008, 10:57
Posts: 69
Location: India
Post Exercise [10.04]
The general polynomial in two variables can be written as,
f(x,y)=\sum{a_{ij}x^{i}y^{j}}where summation takes place over indices i,j.
\frac{\partial{f}}{\partial{x}}=\sum{ia_{ij}x^{i-1}y^{j}}

\frac{\partial{f}}{\partial{y}}=\sum{ja_{ij}x^{i}y^{j-1}}

\frac{\partial^{2}{f}}{\partial{y}\partial{x}}=\frac{\partial}{\partial{y}}(\frac{\partial{f}}{\partial{x}})=\sum{ija_{ij}x^{i-1}y^{j-1}}

\frac{\partial^{2}{f}}{\partial{x}\partial{y}}=\frac{\partial}{\partial{x}}(\frac{\partial{f}}{\partial{y}})=\sum{ija_{ij}x^{i-1}y^{j-1}}

Hence,
\frac{\partial^{2}{f}}{\partial{x}\partial{y}}=\frac{\partial^{2}{f}}{\partial{y}\partial{x}}


23 May 2008, 12:02
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