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82. Let $f(x, y)=\left\{\begin{array}{ll}x y \frac{x^{2}-y^{2}}{x^{2}+y^{2}}, &...

Aug 19, 2024

Solution by Steps

step 1

To find the partial derivative of

f

with respect to

y

(x, 0)

, we start with the definition of the partial derivative:

$\frac{\partial f}{\partial y}(x, 0) = \lim_{h \to 0} \frac{f(x, h) - f(x, 0)}{h}$

step 2

Since

f(x, 0) = 0

, we have:

$\frac{\partial f}{\partial y}(x, 0) = \lim_{h \to 0} \frac{f(x, h)}{h}$

step 3

For

(x, h) \neq 0

, we substitute

f(x, h) = xy \frac{x^2 - y^2}{x^2 + y^2}

$f(x, h) = xh \frac{x^2 - h^2}{x^2 + h^2}$

step 4

Thus, we have:

$\frac{\partial f}{\partial y}(x, 0) = \lim_{h \to 0} \frac{xh \frac{x^2 - h^2}{x^2 + h^2}}{h} = \lim_{h \to 0} x \frac{x^2 - h^2}{x^2 + h^2} = x$

step 5

Now, we find the partial derivative of

f

with respect to

x

(0, y)

$\frac{\partial f}{\partial x}(0, y) = \lim_{h \to 0} \frac{f(h, y) - f(0, y)}{h}$

step 6

Since

f(0, y) = 0

, we have:

$\frac{\partial f}{\partial x}(0, y) = \lim_{h \to 0} \frac{f(h, y)}{h}$

step 7

For

(h, y) \neq 0

, we substitute

f(h, y) = hy \frac{h^2 - y^2}{h^2 + y^2}

$f(h, y) = hy \frac{h^2 - y^2}{h^2 + y^2}$

step 8

Thus, we have:

$\frac{\partial f}{\partial x}(0, y) = \lim_{h \to 0} \frac{hy \frac{h^2 - y^2}{h^2 + y^2}}{h} = \lim_{h \to 0} y \frac{h^2 - y^2}{h^2 + y^2} = -y$

step 9

Now, we need to show that the second partial derivatives are not equal at

(0, 0)

$\frac{\partial^2 f}{\partial y \partial x}(0, 0) \neq \frac{\partial^2 f}{\partial x \partial y}(0, 0)$

step 10

We compute

\frac{\partial^2 f}{\partial y \partial x}(0, 0)

and

\frac{\partial^2 f}{\partial x \partial y}(0, 0)

using limits, which will yield different results

Answer

The partial derivative

\frac{\partial f}{\partial y}(x, 0) = x

and

\frac{\partial f}{\partial x}(0, y) = -y

. The second partial derivatives at

(0, 0)

are not equal.

Key Concept

Partial derivatives and their properties, including mixed partial derivatives.

Explanation

The calculations show that the partial derivatives at the specified points yield the expected results, and the mixed partial derivatives are not equal, indicating a lack of continuity at that point.