Clairaut’s Theorem of Partial Derivatives
suppose that f is defined on a disc D that contains the point (a,b). If both fxy and fyx are continuous, then fxy(a,b)=fyx(a,b)
Clairaut’s theorem states that for a function of two variables that has continuous second partial derivatives, the order of differentiation in computing the mixed partial derivatives does not matter. In other words, it says that if a function f has continuous second partial derivatives, then the second partial derivatives fxy and fyx are equal. Symbolically;
fxy = fyx
This theorem is a useful tool in mathematics and physics, where functions of two or more variables are common. It can simplify the process of computing partial derivatives, and it helps ensure that the results obtained are accurate. Additionally, Clairaut’s theorem is also the foundation of the Schwarz’s theorem, which extends this idea of equality of partial derivatives to the case of functions of more than two variables.
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