Problem Set 4

Consider

$\begin{cases} x^2 - y\cos(uv) = v \\ x^2 + y^2 - \sin(uv) = \tfrac{4}{\pi} u \end{cases}$

(a) Explain why near $(x, y, u, v) = (1, 1, \pi/2, 0)$, the system defines $x = g_1(u, v)$, $y = g_2(u, v)$.

(b) Find $\tfrac{\partial g_1}{\partial u}(\pi/2, 0)$ and $\tfrac{\partial g_2}{\partial u}(\pi/2, 0)$.

(c) Let $\gamma(u, v) = g_1(u, v)^4 + g_2(u, v)^4$. Find $\tfrac{\partial \gamma}{\partial u}(\pi/2, 0)$.

With tables providing values of $f_1, f_2, g$ and their partials at $(0,0), (0,1), (1,0)$:

(a) Explain why $g$ is differentiable at $(0, 0)$ and find $Dg(0, 0)$.

(b) Let $\varphi(x, y) = g(\vec f(x, y))$. Find $\nabla \varphi(0, 1)$ and $D\varphi(0, 1)$.

Find second-order Taylor expansions near $\vec a$, with an explicit bound $|R_{\vec a, 2}(\vec h)| \leq \tfrac{n^3 M}{6}\|\vec h\|_\infty^3$, and determine how small $\|\vec h\|$ must be so $|R_{\vec a, 2}(\vec h)| < 10^{-4}$.

(a) $f(x, y) = \sin(2x) + \cos(x + y)$ near $(0, 0)$.

(b) $f(x, y, z) = e^{-(x^2 + y^2 + z^2)}$ near $(0, 0, 0)$.

Find all maxima and minima of:

(a) $f_1(x, y) = x^3 + xy + y^3$.

(b) $f_2(x, y) = \dfrac{xy}{x^2 + y^2 + 1}$.

(c) $f_3(x, y, z) = x^6 + y^6 + z^6$.

(d) $f_4(x_1, \dots, x_n) = \sum_{i,j} a_i a_j x_i x_j$, where $\vec a \neq \vec 0$.

Find maxima and minima subject to $4x^2 + y^2 = 4y$:

(a) $f_1(x, y) = x^2 y^2 - 4 x^2 y - \tfrac{1}{4} x^4$.

(b) $f_2(x, y) = -4xy + xy^2 - \tfrac{1}{3} x^3$.

Show the set of Liouville numbers has zero Lebesgue measure.