Please note someone attempted to answer the question, however, the person only stated the Greens' th. I would appreciate the full proof.

To better understand how conservation equations deal with these often inevitable discontinuities, we study the Riemann Problem. This is a conser- vation equation initial value problem with: 20(L) = U1 I 0 In class, we have shown that Su r st where s = (ui + ur)/2 is a weak solution when u > Ur. This was calculated by integrating over three regions of our domain. We can also verify this is a weak solution by integrating over two regions (u = , u = Ur), by applying Green's Theorem. Exercise 1.3. Show that u is a weak solution to this Riemann Problem via Green's Theorem. Hint: OzU1 = (qui)since uz is constant, we use Green's Theorem to reframe S S 0{U1 + $f(ui). The same goes for the right region. To better understand how conservation equations deal with these often inevitable discontinuities, we study the Riemann Problem. This is a conser- vation equation initial value problem with: 20(L) = U1 I 0 In class, we have shown that Su r st where s = (ui + ur)/2 is a weak solution when u > Ur. This was calculated by integrating over three regions of our domain. We can also verify this is a weak solution by integrating over two regions (u = , u = Ur), by applying Green's Theorem. Exercise 1.3. Show that u is a weak solution to this Riemann Problem via Green's Theorem. Hint: OzU1 = (qui)since uz is constant, we use Green's Theorem to reframe S S 0{U1 + $f(ui). The same goes for the right region