Clairaut's Theorem states that if a function has continuous second partial derivatives, then the order of differentiation does not matter; that is, the mixed partial derivatives are equal. This theorem is important in understanding how to compute higher-order derivatives and ensures that we can interchange the order of differentiation for functions with certain smoothness conditions, linking it to both the definition of partial derivatives and higher-order partial derivatives.
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Clairaut's Theorem applies specifically when the second partial derivatives of a function are continuous at a point or over a region.
The theorem can be expressed mathematically as $$ rac{{\partial^2 f}}{{\partial x \partial y}} = rac{{\partial^2 f}}{{\partial y \partial x}}$$ for a function $$f$$ of two variables.
The significance of Clairaut's Theorem is that it simplifies calculations in multivariable calculus by allowing the interchange of the order of differentiation without worrying about the results being different.
While Clairaut's Theorem assumes continuity of second partial derivatives, failure of this condition may lead to mixed partial derivatives being unequal.
Understanding Clairaut's Theorem helps in analyzing the properties of functions that arise in physics and engineering, where multivariable functions frequently occur.
Review Questions
How does Clairaut's Theorem help in simplifying the computation of mixed partial derivatives?
Clairaut's Theorem states that if a function has continuous second partial derivatives, you can differentiate in any order without affecting the outcome. This means that when computing mixed partial derivatives, you can choose whether to differentiate with respect to one variable first or another and still arrive at the same result. This property streamlines calculations significantly, especially in problems involving complex multivariable functions.
What conditions must be satisfied for Clairaut's Theorem to hold true, and why are these conditions important?
For Clairaut's Theorem to be valid, the second partial derivatives of the function must be continuous in a neighborhood around the point of interest. This condition is crucial because if the second partials are not continuous, it is possible for the mixed partial derivatives to differ. Ensuring continuity allows us to confidently interchange the order of differentiation, which is essential for accurate computations in multivariable calculus.
Evaluate a function $$f(x,y) = x^2y + y^3$$ at point (1, 2) and demonstrate how Clairaut's Theorem applies when finding the mixed partial derivatives.
To evaluate the mixed partial derivatives for $$f(x,y) = x^2y + y^3$$ at point (1, 2), we first compute the first-order partial derivatives: $$f_x = 2xy$$ and $$f_y = x^2 + 3y^2$$. Then we find the second-order mixed derivatives: $$f_{xy} = 2y$$ and $$f_{yx} = 2y$$ as well. Since both mixed partials are equal at (1, 2), Clairaut's Theorem holds here, confirming that we can interchange differentiation without loss of accuracy due to the continuity of the second derivatives.