Partial Derivatives Equality Theorem for Mixed Partial Derivatives in Calculus 3

What is the Equality of Mixed Partial Derivatives Theorem in Mathematics?

The Equality of Mixed Partial Derivatives Theorem, also known as Clairaut's Theorem, states that if the mixed partial derivatives of a function are continuous, then these mixed partial derivatives are equal. This theorem provides a condition under which the order of differentiation does not affect the result.

Q&A:

Question: What is the statement of Clairaut's Theorem (Equality of Mixed Partial Derivatives Theorem)?

Answer: Clairaut's Theorem states that if a function f(x, y) has continuous second-order partial derivatives, then the mixed partial derivatives are equal. Formally, if f is a function defined on an open subset of R^2 and the second partial derivatives fxy and fyx are continuous, then:
fxy = fyx

Question: What are mixed partial derivatives?

Answer: Mixed partial derivatives are the derivatives obtained by differentiating a function of more than one variable with respect to different variables in succession. For instance, if we have a function f(x, y), the mixed partial derivative fxy is the derivative of f first with respect to x and then with respect to y.

Question: Can you provide an example to illustrate Clairaut's Theorem?

Answer: Certainly! Consider the function f(x, y) = x^2y + 3xy^2. Let us compute the first and second-order partial derivatives to see this theorem in action.
1. Find the first partial derivative with respect to x:
fx = 2xy + 3y^2
2. Now, differentiate fx with respect to y to get fxy:
fxy = 2x + 6y

3. Similarly, find the first partial derivative with respect to y:
fy = x^2 + 6xy
4. Differentiate fy with respect to x to get fyx:
fyx = 2x + 6y

Now, according to Clairaut's Theorem:
fxy = fyx
That is, both mixed partial derivatives fxy and fyx are equal, which are 2x + 6y in this case.

Question: Why is the continuity of the mixed partial derivatives important for Clairaut's Theorem?

Answer: Continuity of the mixed partial derivatives is crucial because it guarantees the equivalence when we interchange the order of differentiation. If the mixed partial derivatives are not continuous, the interchange of partial differentiation may not yield the same result, and therefore, Clairaut's Theorem might not hold.

Question: Can Clairaut's Theorem be applied to higher dimensions and more variables?

Answer: Yes, Clairaut's Theorem can be extended to functions of more than two variables and higher-order mixed partial derivatives. The same principle applies: if all the mixed partial derivatives of a certain order are continuous, they will be equal regardless of the order in which the differentiations are performed.

Let’s consider a function f(x1, x2, ..., xn) with continuous higher-order partial derivatives. For example, for three variables:
f(x, y, z), we have:
fx, yz = fz, yx = fy, xz = fxy z

These mixed partial derivatives will be equal if they are continuous.

Calculus 3: Partial Derivatives Equality Theorem for Mixed Partial Derivatives

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