MATH528

MATH528 Lesson02: Exact differential forms

Definition. For any functions $M, N : ℝ^{2} \to ℝ$ , $M (x, y) d x + N (x, y) d y$ is a differential form.

Definition. The differential form $M (x, y) d x + N (x, y) d y$ is said to be exact if $\exists u \in C^{1} (ℝ^{2})$ such that

$d u = M (x, y) d x + N (x, y) d y \Rightarrow \frac{\partial u}{\partial x} = M (x, y), \frac{\partial u}{\partial y} = N (x, y)$

Definition. $M (x, y) d x + N (x, y) d y = 0$ is called a differential equation.

If $M (x, y) d x + N (x, y) d y$ is exact, the associated differential equation has the solution
$u (x, y) = c,$
an implicit solution for the function $y (x)$ .
For an exact differential form with $u \in C^{2} (ℝ^{2})$ we have $\frac{\partial M}{\partial y} = \frac{\partial^{2} u}{\partial y \partial x} = \frac{\partial^{2} u}{\partial x \partial y} = \frac{\partial N}{\partial x} \Rightarrow \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

Example. $\cos (x + y) d x + (3 y^{2} + 2 y + \cos (x + y)) d y = 0$ .

In[33]:= Dform = Cos[x+y] dx + (3y^2+2y+Cos[x+y]) dy; Deq = Dform == 0

$dx \cos (x + y) + dy (\cos (x + y) + 3 y^{2} + 2 y) = 0$

In[34]:= {Mf[x_,y_],Nf[x_,y_]} = {Coefficient[Dform,dx],Coefficient[Dform,dy]}

${\cos (x + y), \cos (x + y) + 3 y^{2} + 2 y}$

In[35]:= D[Mf[x,y],y] == D[Nf[x,y],x]

$True$