Question

Let \(f(x, y)=x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}} \quad\) if \(\quad(x, y) \neq(0,0)\) and \(f(0,0)=0\) Show that \(f_{x y}(0,0) \neq f_{y x}(0,0)\) by completing the following steps: (a) Show that \(f_{x}(0, y)=\lim _{h \rightarrow 0} \frac{f(0+h, y)-f(0, y)}{h}=-y\) for all \(y\). (b) Similarly, show that \(f_{y}(x, 0)=x\) for all \(x\). (c) Show that \(f_{y x}(0,0)=\lim _{h \rightarrow 0} \frac{f_{y}(0+h, 0)-f_{y}(0,0)}{h}=1\). (d) Similarly, show that \(f_{x y}(0,0)=-1\).

Short answer

Yes, \(f_{xy}(0,0) \neq f_{yx}(0,0)\) since \(f_{xy}(0,0) = -1\) and \(f_{yx}(0,0) = 1\).

Step-by-step solution

Differentiate with respect to x at (0, y)

To find \(f_x(0, y)\), we need to use the limit definition of partial derivatives. We evaluate:\[f_x(0, y) = \lim_{h \to 0} \frac{f(0+h, y) - f(0, y)}{h}.\]Given \(f(x, y) = x y \frac{x^2 - y^2}{x^2 + y^2}\), we substitute in \((h, y)\):\[f(h, y) = hy \frac{h^2 - y^2}{h^2 + y^2}.\]Since \(f(0, y) = 0\), we have:\[f_x(0, y) = \lim_{h \to 0} \frac{hy \frac{h^2 - y^2}{h^2 + y^2}}{h} = \lim_{h \to 0} y \frac{h^2 - y^2}{h^2 + y^2}.\]Examining the expression as \(h \to 0\):\[f_x(0, y) = \lim_{h \to 0} \left(y \frac{0 - y^2}{0 + y^2}\right) = -y.\]Thus, \(f_x(0, y) = -y\).

Differentiate with respect to y at (x, 0)

Next, we find \(f_y(x, 0)\). Again, we use the limit definition:\[f_y(x, 0) = \lim_{k \to 0} \frac{f(x, 0+k) - f(x, 0)}{k}.\]Substituting in \((x, k)\):\[f(x, k) = xk \frac{x^2 - k^2}{x^2 + k^2}.\]Since \(f(x, 0) = 0\), we have:\[f_y(x, 0) = \lim_{k \to 0} \frac{xk \frac{x^2 - k^2}{x^2 + k^2}}{k} = \lim_{k \to 0} x \frac{x^2 - k^2}{x^2 + k^2}.\]Examining this expression as \(k \to 0\):\[f_y(x, 0) = \lim_{k \to 0} \left(x \frac{x^2 - 0}{x^2 + 0}\right) = x.\]Thus, \(f_y(x, 0) = x\).

Evaluate the mixed derivative f_yx(0,0)

To calculate \(f_{yx}(0,0)\), we differentiate \(f_y(x, 0)\) with respect to \(x\):\[f_{yx}(0, 0) = \lim_{h \to 0} \frac{f_y(0+h, 0) - f_y(0, 0)}{h}.\]We substitute what we found earlier for \(f_y\):\[f_y(h, 0) = h.\]Thus:\[f_{yx}(0, 0) = \lim_{h \to 0} \frac{h - 0}{h} = 1.\]Hence, \(f_{yx}(0, 0) = 1\).

Evaluate the mixed derivative f_xy(0,0)

Now, we calculate \(f_{xy}(0, 0)\), differentiating \(f_x(0, y)\) with respect to \(y\):\[f_{xy}(0, 0) = \lim_{k \to 0} \frac{f_x(0, 0+k) - f_x(0, 0)}{k}.\]With \(f_x(0, y) = -y\), we use:\[f_x(0, k) = -k.\]Therefore:\[f_{xy}(0, 0) = \lim_{k \to 0} \frac{-k - 0}{k} = -1.\]Thus, \(f_{xy}(0, 0) = -1\).

Key concepts

Partial Derivatives

Partial derivatives are a fundamental concept when working with multivariable functions. They allow us to examine how a function changes with respect to one variable, while keeping other variables constant. This is particularly useful for functions with more than one variable, like in our exercise where the function is defined as \(f(x,y)\).

Mathematically, a partial derivative of a function \(f(x,y)\) with respect to \(x\), noted as \(f_x\), is obtained by differentiating \(f\) with respect to \(x\) and treating \(y\) as a constant. Similarly, the partial derivative with respect to \(y\), noted as \(f_y\), is found by differentiating with respect to \(y\), treating \(x\) as a constant.

• Partial derivatives help in understanding the rate of change in a multivariable context.
• They are used extensively in calculus, physics, and engineering to understand how systems change.
• An important point to note is that while taking a partial derivative, other variables are considered fixed.

Mixed Derivatives

Mixed derivatives involve differentiating a function with respect to multiple variables in sequence. In multivariable calculus, sometimes we differentiate first with respect to one variable and then with respect to another. These are called mixed partial derivatives. For example, \(f_{xy}\) indicates that we first find the partial derivative with respect to \(x\) and then differentiate that result with respect to \(y\).

In an ideal world, the order of differentiation wouldn't matter due to Clairaut's theorem stating \(f_{xy} = f_{yx}\) if the function's second derivatives are continuous. However, in exercises like the one provided, we observe that this doesn't always hold, particularly if discontinuities or non-standard functions are involved.

• Mixed derivatives help us describe complex changes in multivariable contexts.
• The order of differentiation may not be interchangeable if the function lacks continuity.
• This exercise exemplifies a case where \(f_{xy} eq f_{yx}\).

Limit Definition

The limit definition is a foundational principle in calculus used to define concepts such as derivatives. When working with partial derivatives, we apply the limit definition to find the rate of change of a function with respect to one variable. The limit definition is mathematical and involves evaluating the function's behavior as it approaches a certain value, usually zero.

The process usually involves taking the difference quotient and then finding the limit as the increment approaches zero. In our exercise, the limit definition helps us calculate both partial and mixed derivatives. For instance, to find \(f_x(0, y)\), we use the limit definition:

• Examining small changes in variables aids in understanding function behavior.
• Limits form the theoretical backbone of defining both ordinary and partial derivatives.
• Calculating derivatives using limits can sometimes reveal unusual properties, such as non-equal mixed derivatives.

Multivariable Functions

Multivariable functions consider multiple input variables, giving us a broad understanding of mathematical systems. Instead of only one independent variable, these functions handle two or more, such as \(f(x, y)\). Understanding changes in such functions involves considering how each variable individually affects the outcome and its interplay with other variables.

These functions are essential in fields like optimization, economics, and engineering, where systems with multiple influencing factors are common. Grasping the behavior of such functions is crucial for real-world problem solving.

• They allow us to model systems with several influencing factors.
• Analyzing them often includes calculating partial and mixed derivatives.
• Understanding these functions is key for applying calculus in practical scenarios.