Prove that if f is a function of two variables and all the partial derivatives of f up to the fo

Prove that if f is a function of two variables and all the...

Key Concepts

Mixed Partial Derivatives

Mixed partial derivatives are the derivatives obtained by differentiating a function with respect to different variables in succession. Their equality, under certain conditions such as continuous higher order derivatives, is a central concept in the study of differential calculus in multiple dimensions and is essential for many theoretical and practical applications.

Open Set in the Domain

The condition that the function is defined on an open disk implies that there exists a neighborhood around every point in the domain where the required partial derivatives exist and are continuous. This property is important because many results in multivariable calculus, including the equality of mixed partial derivatives, rely on the behavior of functions in open sets.

Clairaut's Theorem

Clairaut's theorem states that if the mixed partial derivatives of a function are continuous in a neighborhood of a point, then these derivatives are equal regardless of the order of differentiation. This theorem is fundamental in multivariable calculus and guarantees that under the given continuity conditions, D1122 f equals D2121 f.

Continuity of Higher Order Partial Derivatives

The continuity of higher order partial derivatives is a crucial condition that allows one to interchange the order of differentiation. When all partial derivatives up to the fourth order are continuous, it validates the use of iterative applications of Clairaut's theorem, ensuring that mixed derivative operations are commutative.

Transcript

00:01 For this problem, we are going to apply a clarote's theorem to show three things.

00:06 We want to show that f, x, y, y, equals f, y, x, y, x, y, x, y, equals f y, x, y, x.

00:15 In other words, we have three third partial derivatives.

00:21 We have taken it twice with respect to y and once with respect to x.

00:25 We want to show that it doesn't matter what order we do this in, as long as i do it once by x and twice by y, i should get the same result.

00:34 Now, let's review clear oats theorem.

00:37 Clareote's theorem says if i have second partial derivatives, partial with respect to x and then y, or if i go y and then x, it doesn't matter.

00:48 As long as i do it once with each, i get the same result.

00:51 So we're going to show that we can use this idea here and expand it to a third partial derivative.

00:58 We're going to do this in two pieces.

01:01 First, let's take a look at the right -hand side.

01:05 So i'm going to write, i'm actually going to copy clarote's theorem down for us.

01:11 The partial derivative, the respect to x and then y equals that same partial derivative, first taken for y and then x.

01:18 So those are the same.

01:20 So i'm just going to take the partial derivative with respect to y to each side.

01:25 So i have f sub x, y, again, partial derivative with y, partial derivative with y.

01:33 I've just taken the same partial derivative on both sides since what i started with was equal...

Please give Ace some feedback