Higher-order partial differentiation

-


Given the function \(z = f(x,y)\), suppose that the partial derivatives \(f_x(x,y)\) and \(f_y(x,y)\) exist, and they also have partial derivatives. For example, the partial derivative of \(f_x(x,y)\) with respect to \(y\),

\[(f_x)_y(x,y) = \frac{\partial}{\partial y}\left(\frac{\partial f}{\partial x}\right)(x,y) = \frac{\partial}{\partial y}\left(\frac{\partial z}{\partial x}\right)(x,y),\]

is denoted as \(\frac{\partial^2f}{\partial y\partial x}(x,y)\) or \(f_{xy}(x,y)\). Similarly, we may define \(f_{xx}(x,y)\), \(f_{yx}(x,y)\), \(f_{yy}(x,y)\). These are called second partial derivatives.

Example. Let \(f(x,y) = \log(x^2 + xy + 2y^2)\). Then,

\[\begin{eqnarray*} f_x(x,y) &=& \frac{2x + y}{x^2 + xy + 2y^2},\\ f_y(x,y) &=& \frac{x + 4y}{x^2 + xy + 2y^2},\\ f_{xx}(x,y) &=& -\frac{2x^2 + 2xy - 3y^2}{(x^2 + xy + 2y^2)^2},\\ f_{xy}(x,y) &=& -\frac{x^2 + 8xy + 2y^2}{(x^2 + xy + 2y^2)^2},\\ f_{yx}(x,y) &=& -\frac{x^2 + 8xy + 2y^2}{(x^2 + xy + 2y^2)^2},\\ f_{yy}(x,y) &=& \frac{3x^2 - 4xy - 8y^2}{(x^2 + xy + 2y^2)^2}. \end{eqnarray*}\] □

In the above example, we have \(f_{xy}(x,y) = f_{yx}(x,y)\). Is this generally the case? Short answer: Not always. In general, we have the following result.

Theorem (Changing the order of partial differentiation)

Suppose the function \(f(x,y)\) on an open region \(U\) has second derivatives \(f_{xy}(x,y)\) and \(f_{yx}(x,y)\) both of which are continuous. Then \(f_{xy}(x,y) = f_{yx}(x,y)\).

Proof. We show that \(f_{xy}(a,b) = f_{yx}(a,b)\) for every \((a, b) \in U\). 

Take a sufficiently small \(\delta > 0\) so that \(|x - a| < \delta\) and \(|y - b| < \delta\) imply \((x, y) \in U\) (Note: This is always possible because \(U\) is open). For all \(h,k\) such that \(0 < |h| < \delta\) and \(0 < |k| < \delta\), let

\[F(h,k) = f(a+h, b+k) - f(a+h, b) - f(a,b+k) + f(a,b).\]

Define the univariate function \(u(y)\) of \(y\) by

\[u(y) = f(a + h, y) - f(a,y).\]

Then we can write

\[F(h, k) = u(b+k) - u(b).\]

\(u(y)\) is differentiable with respect to \(y\) and \(u'(y) = f_y(a+h,y) - f_y(a,y)\). By the Mean Value Theorem, there exists \(\theta \in (0, 1)\) such that

\[F(h,k) = u'(b+\theta k)k = k[f_y(a+h, b+\theta k) - f_y(a, b+\theta k)].\]

  Next, consider the univariate function of \(x\): \(f_y(x, b + \theta k)\). Again, by the Mean Value Theorem, there exists \(\eta \in (0, 1)\) such that

\[F(h,k) = hkf_{yx}(a + \eta h, b + \theta k).\tag{Eq:Fhk1}\]

If we repeat the above process by swapping the roles of \(x\) and \(y\), we can show (exercise!) that there exist \(\theta', \eta' \in (0, 1)\) such that

\[F(h,k) = hkf_{xy}(a + \eta' h, b + \theta' k).\tag{Eq:Fhk2}\]

From Eqs. (Eq:Fhk1) and (Eq:Fhk2), it follows that

\[f_{yx}(a + \eta h, b + \theta k) = f_{xy}(a + \eta' h, b + \theta' k).\]

By the continuity of \(f_{xy}(x,y)\) and \(f_{yx}(x,y)\), as \((h,k) \to (0,0)\),

\[f_{yx}(a,b) = f_{xy}(a,b).\] ■

The second derivatives of the function \(f(x,y)\) may have derivatives. For example,

\[f_{yxy}(x,y) = \frac{\partial}{\partial y}f_{yx}(x,y).\]

If \(f_{xy}(x,y)\) and \(f_{yx}(x,y)\) are continuous, by the above theorem, \(f_{xy}(x,y) = f_{yx}(x,y)\). It follows that \(f_{xyy}(x,y) = f_{yxy}(x,y)\), that is,

\[\frac{\partial^3f}{\partial y^2\partial x} = \frac{\partial^3f}{\partial y\partial x\partial y}.\]

These are called third derivatives. 

Remark. We usually write

\[\frac{\partial^2 f}{\partial x^2}(x,y)\]

for \(f_{xx}(x,y)\), rather than

\[\frac{\partial^2 f}{\partial x\partial x}(x,y).\]

□

Definition (Functions of class \(C^n\))

Let \(f(x,y)\) be a function on an open region \(U\) and \(n\) be a non-negative integer.

  1. \(f(x,y)\) is said to be \(n\)-times continuously differentiable or of class \(C^n\) if it has all the partial derivatives up to the \(n\)-th order on \(U\), all of which are continuous.
  2. \(f(x,y)\) is said to be infinitely differentiable, smooth, or of class \(C^{\infty}\) if it has the partial derivatives of all orders on \(U\), all of which are continuous.
In general, the derivatives of a function of class \(C^n\) are determined by the number of partial differentiation with respect to \(x\) and \(y\) and are independent of the order of differentiation.

Example. If \(f(x,y)\) is of class \(C^3\), then \(f_{xxy}(x,y) = f_{xyx}(x,y) = f_{yxx}(x,y)\).



Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

-
Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

-
Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Newton's method

-
Image
Newton's method (the Newton-Raphson method) is a very powerful numerical method for solving nonlinear equations.  Suppose we'd like to solve a non-linear equation \(f(x) = 0\), where \(f(x)\) is a (twice) differentiable non-linear function. Newton's method generates a sequence of numbers \(c_1, c_2, c_3, \cdots\) that converges to a solution of the equation. That is, if \(\alpha\) is a solution (i.e., \(f(\alpha) = 0\)) then,  \[\lim_{n\to\infty}c_n = \alpha,\] and this sequence \(\{c_n\}\) is generated by a series of linear approximations of the function \(f(x)\). Theorem (Newton's method) Let \(f(x)\) be a function that is twice differentiable on an open interval \(I\) that contains the closed interval \([a, b]\) (i.e., \([a,b]\subset I\)) and satisfy the following conditions: \(f(a) < 0\) and \(f(b) > 0\); For all \(x\in [a, b]\), \(f'(x) > 0\) and \(f''(x) > 0\). Let us define the sequence \(\{c_n\}\) by \[ \begin{eqnarray} c_1 &
Read more