1. Let $V = \R^2$, and let $v = (3,2)$ in the Cartesian basis of $\R^2$. Now, we choose another basis as follows $$ e_1 = (2,1), \quad e_2 = (0,1) $$. Expand $v$ in terms of $e_1, e_2$.

2. Let $V = \{a + bt + ct^2 \mid a, b, c \in \R\} $ be the space of polynomials of degree at most 2. Let $f_1(t), f_2(t), f_3(t)$ be three elements in $V$, given as follows $$ f_1(2) = 1, \quad f_1(3) = 0, \quad f_1(5) = 0 $$ $$ f_2(2) = 0, \quad f_2(3) = 1, \quad f_2(5) = 0 $$ $$ f_3(2) = 0, \quad f_3(3) = 0, \quad f_3(5) = 1 $$ Then

• Show that $f_1, f_2, f_3$ forms a basis of $V$.
• Expand $f = 3t^2 + 1$ in terms of this basis.

3. Consider the new coordiante $(u,v)$ on $\R^2$, related to $(x,y)$ by $$ x = \cosh(u) \cos(v), \quad y = \sinh(u) \sin(v) $$

• Find the metric tensor $g$ in terms of $u,v$ (or equivalently, the line element $ds^2$),
• The volume element.
• The expansion of $\d_u$ and $\d_v$ using $\d_x$ and $\d_y$ (or equivalently, the line element $ds^2$).
• (bonus) If a function on $\R^2$ is given by $f(u,v) = 2u + 3v$, find its gradient expressed using basis vectors $\d_u, \d_v$.
• (bonus) Find the divergence of the vector field $V = \d_u$.

You may use the formula $$ grad(f) = g^{ij}\, \d_i f \,\d_j $$ and $$ div(V^i \d_i) = \frac{1}{\sqrt{g}} \d_i(\sqrt{g} V^i) $$ where we used Einstein summation convention, and $\d_i = \frac{\d}{\d u_i}$, $u_1 = u, u_2 = v$.