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\centerline{{\sc Problem set 1}}
\centerline{Math 207B, Winter 2016}
\centerline{Due: Fri., Jan. 22}


\bigskip\noindent
{\bf 1.} Define $f:\Rl^2 \to \Rl$ by $f(0,0) = 0$ and
\[
f(x,y) = \frac{xy^3}{x^2 + y^6}\qquad \mbox{if $(x,y)\ne (0,0)$}.
\]
(a) Show that the directional derivatives of $f$ at $(0,0)$ exist in every direction.
What is its G\^ateaux derivative at $(0,0)$?

\smallskip\noindent
(b) Show that $f$ is not Fr\'echet differentiable at $(0,0)$. (\textsc{Hint}. A Fr\'echet differentiable function must be continuous.)

\bigskip\noindent
{\bf 2.} Define $f, g : \Rl^2 \to \Rl$ by
\[
f(x,y) = x^2 + y^2,\qquad g(x,y) = (y-1)^3 - x^2.
\]
Find the minimum value of $f(x,y)$ subject to the constraint $g(x,y)=0$. Show that there does not exist any constant $\lambda$ such that
$\nabla f= \lambda \nabla g$ at some point $(x,y)\in \Rl^2$. Why does the method of Lagrange multipliers fail in this example?


\bigskip\noindent
{\bf 3.} Derive the Euler-Lagrange equation for a functional of the form
\[
J(u) = \int_a^b F(x,u,u',u'')\, dx.
\]
What are the natural boundary conditions for this functional?

\bigskip\noindent
{\bf 4.} A curve $y = u(x)$ with $a\le x \le b$, $u(x) > 0$, and $u(a) = u_0$, $u(b) = u_1$ is rotated about the $x$-axis. Find the
curve that minimizes the area of the surface of revolution,
\[
J(u) = \int_a^b u\sqrt{1 + (u')^2}\, dx.
\]


\newpage\noindent
{\bf 5.} Let $X$ be the space of smooth functions $u :[0,1]\to \Rl$ such that $u(0) = 0$, $u(1) = 0$. Define functionals $J,K : X \to \Rl$ by
\[
J(u) = \frac{1}{2} \int_0^1 (u')^2\, dx,\qquad K(u) = \frac{1}{2} \int_0^1 u^2\, dx.
\]
(a) Introduce a Lagrange multiplier and write down the Euler-Lagrange equation for extremals in $X$ of the functional $J(u)$ subject to the constraint $K(u) = 1$.

\smallskip\noindent
(b) Solve the eigenvalue problem in (a) and find all of the extremals. Which one minimizes $J(u)$?



\bigskip\noindent
{\bf 6.} (a) Make a change of variable $x = \phi(t)$, $v(t) = u\left(\phi(t)\right)$,
where $\phi'(t) > 0$, in the functional
\[
J(u) = \int_a^b F(x,u,u')\, dx.
\]
Show that $J(u)=K(v)$ where $K(v)$ has the form
\[
K(v) = \int_{c}^d G(t,v,v')\, dt
\]
and express $G$ in terms of $F$ and $\phi$.

\smallskip\noindent
(b) Show that the Euler-Lagrange equation for $K(v)$ is the same as what you get by changing variables in the Euler-Lagrange equation for $J(u)$.


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