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Problem 129

(a) Let \(\mathrm{f}: \mathrm{R}^{2} \rightarrow \mathrm{R}\) be defined by $f(x, y)=2 x y\left\\{\left(x^{2}-y^{2}\right) /\left(x^{2}+y^{2}\right)\right\\}, x^{2}+y^{2} \neq 0$ and \(=0, \quad \mathrm{x}=\mathrm{y}=0\). Show that $\left(\partial^{2} \mathbf{f} / \partial \mathrm{x} \partial \mathrm{y}\right) \neq\left(\partial^{2} \mathrm{f} / \partial \mathrm{x} \partial \mathrm{y}\right)$ and explain why. (b) Does there exist a function \(\mathrm{f}\) with continuous second partial derivatives (i.e., an element of \(\mathrm{C}^{2}\) ) such that of $/ \partial \mathrm{x}=\mathrm{x}^{2}$ and \partialf \(/ \partial \mathrm{y}=\mathrm{xy}\) ?

Expert verified

In summary:
a) The second mixed partial derivatives of the given function are not globally equal, as they are unequal for \((x, y) \neq (0, 0)\). This is because the function is not differentiable at the origin, so it does not satisfy the necessary requirements for the equality of mixed partial derivatives.
b) There does not exist a function \(f\) with continuous second partial derivatives such that our given conditions \(\frac{\partial f}{\partial x} = x^2\) and \(\frac{\partial f}{\partial y} = xy\) are satisfied, as a function solely dependent on \(y\) cannot be equal to \(xy\).

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Chapter 5

Using Taylor's Theorem, approximate \(\sqrt{40}\) to three decimal places.

Chapter 5

(a) State and prove Euler's Theorem on positively homogeneous functions of two variables. (b) Let \(\mathrm{F}(\mathrm{x}, \mathrm{y})\) be positively homogeneous of degree 2 and $\mathrm{u}=\mathrm{r}^{\mathrm{m}} \mathrm{F}(\mathrm{x}, \mathrm{y})\( where \)\mathrm{r}=\left(\mathrm{x}^{2}+\mathrm{y}^{2}\right)^{1 / 2}$. Show that $\left(\partial^{2} \mathrm{u} / \partial \mathrm{x}^{2}\right)+\left(\partial^{2} \mathrm{u} / \partial \mathrm{y}^{2}\right)$ $=\mathrm{r}^{\mathrm{m}}\left\\{\left(\partial^{2} \mathrm{~F} / \partial \mathrm{x}^{2}\right)+\left(\partial^{2} \mathrm{~F} / \partial \mathrm{y}^{2}\right)\right\\}+\mathrm{m}(\mathrm{m}+4) \mathrm{r}^{\mathrm{m}-2} \mathrm{~F}$

Chapter 5

Show that the functions $\mathrm{f}, \mathrm{g} \in \mathrm{C}^{1}(\mathrm{E}), \mathrm{E}\( open in \)\mathrm{R}^{2}$, are functionally dependent (i.e., there exists a function \(\mathrm{F}\) such that \(g=F^{\circ} \mathrm{f}\) ) if det \(J \phi(x, y)=0\) for \(\Phi=(f, g)\) and $(x, y)\( in some neighborhood of \)(a, b)\(, where \)(\partial f / \partial x)(a, b) \neq 0$

Chapter 5

State and prove the Cauchy Mean Value Theorem.

Chapter 5

Give a Taylor expansion of \(f(x, y)=e^{x} \cos y\) on some compact convex domain \(\mathrm{E}\) containing \((0,0)\).

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