Suppose we have two 2D vector fields in $ (x, y) $, $ boldsymbol {f} $ Y $ boldsymbol {g} $, which are defined by two `InterpolatingFunction`

the obtained from `NDSolve`

. I wonder what is the efficient and reasonable method to "compare" and visualize to show which effect represented by the fields is "bigger" than the other in a certain region of $ (x, y) $ airplane?

For two scalars, the proportion of them can reveal their relative magnitude. But is it reasonable to compare the norm of the two vector fields with a relation of $ || boldsymbol {f} || / || boldsymbol {g} || $, which must be in contour $ (x, y) $ airplane? Another challenge is that the fields depend on time, since I also want to show the change of their relative magnitude with time.

Consider the following PDEs about $ u (x, y, t) $ Y $ v (x, y, t) $,

```
L = 4;
sol = NDSolve({D(u(t, x, y), t, t) ==
D(u(t, x, y), x, x) + D(u(t, x, y), y, y) + Sin(u(t, x, y)),
u(t, -L, y) == u(t, L, y), u(t, x, -L) == u(t, x, L),
u(0, x, y) == Exp(-(x^2 + y^2)), Derivative(1, 0, 0)(u)(0, x, y) == 0},
u, {t, 0, 4}, {x, -L, L}, {y, -L, L})
NDSolve({D(v(t,x,y),t,t)==D(v(t,x,y),x,x)+D(v(t,x,y),y,y)/2+(1-v(t,x,y)^2)(1+2v(t,x,y)),
v(0,x,y)==E^-(x^2+y^2),v(t,-L,y)==v(t,L,y),v(t,x,-L)==v(t,x,L),(v^(1,0,0))(0,x,y)==0},
v,{t, 0, 4}, {x, -L, L}, {y, -L, L})
```

and these two vector fields

```
f = u(t, x, y)*Grad(v(t, x, y), {x, y}) + Grad(u(t, x, y), {x, y})
g = Grad(u(t, x, y), {x, y})*u(t, x, y)
```

Thanks in advance.