its coordinate form 8 > > > > < > > > > : @t X (⇠ 1 , ⇠ 2 , t) + HX (X, g 1 ⇠ r⇠ X ) = F(X, ⇢(·, t)), @t⇢X (⇠ 1 , ⇠ 2 , t) 1 p det(g⇠ ) 2 X i=1 @ @⇠i 2 X j=1 p det(g⇠ )⇢X (g 1 ⇠ )ij@qj HX ! = 0, X (⇠ 1 , ⇠ 2 , 1) = FT (X(⇠ 1 , ⇠ 2), ⇢(·, 1)), ⇢X (⇠ 1 , ⇠ 2 , 0) = ⇢0(X(⇠ 1 , ⇠ 2)). I Comparing to the PDE formulation, when F(⇢) ⇢ (x) = F(x, ⇢), FT (⇢) ⇢ (x) = FT (x, ⇢), the equivalent variational formulation is easier to handle min ⇢,m Z 1 0 Z M ⇢(x, t)L ✓ x, m(x, t) ⇢(x, t) ◆ dMxdt + Z 1 0 F(⇢(·, t))dt + FT (⇢(·, 1)), s.t. @t⇢(x, t) + rM · m(x, t) = 0, ⇢(·, 0) = ⇢0. A local coordinate representation of a manifold. 3 34