# isInvolutive -- tests if a distribution is involutive

## Synopsis

• Usage:
isInvolutive(L)
• Inputs:
• Outputs:
• , true if the distribution L is involutive

## Description

If L is a list of vector fields, this routine tests the involutivity of L.

In this example we test the involutivity of two vector fields.

 i1 : X = newField("3*x_0*ax_0+x_1*ax_1") o1 = 3x ax + x ax 0 0 1 1 o1 : DiffAlgField i2 : Y = radial 3 o2 = x ax + x ax + x ax + x ax 0 0 1 1 2 2 3 3 o2 : DiffAlgField i3 : isInvolutive dist {X,Y} o3 = true

In this example we compute a basis of the annihilator of a random projective logarithmic differential 1-form. Then we verify that it is an involutive distribution.

 i4 : w = random logarithmicForm(2,{1,2},"a",Projective => true) 2 2 2 o4 = (120x x + 144x + 306x x + 108x x + 108x )dx + (- 120x - 144x x - 0 1 1 0 2 1 2 2 0 0 0 1 ------------------------------------------------------------------------ 2 2 2 108x x - 264x x - 72x )dx + (- 306x + 264x - 108x x + 72x x )dx 0 2 1 2 2 1 0 1 0 2 1 2 2 o4 : DiffAlgForm i5 : X = newField(2,2,"a") 2 2 2 2 o5 = (a x + a x x + a x + a x x + a x x + a x )ax + (a x + a x x + 0 0 3 0 1 9 1 6 0 2 12 1 2 15 2 0 1 0 4 0 1 ------------------------------------------------------------------------ 2 2 2 2 a x + a x x + a x x + a x )ax + (a x + a x x + a x + a x x + 10 1 7 0 2 13 1 2 16 2 1 2 0 5 0 1 11 1 8 0 2 ------------------------------------------------------------------------ 2 a x x + a x )ax 14 1 2 17 2 2 o5 : DiffAlgField i6 : D = genKer(X_w,X); i7 : #D o7 = 6 i8 : isInvolutive dist D o8 = true