Choose a grid of points on and sketch the tangent vectors of the vector field [ x ₂, x ² ₁] ^T at those points. You may draw these with a computer if you wish. Sketch by hand or use a computer to draw a few integral curves of this vector field. Does this vector field define aregular one-dimensional distribution?

For the vector fields g ₁ = [ x ₁ + x ₂,0] ^T and g ₂ = [0, 1 + x ₂] ^T on , sketch the integral manifolds, or foliation, defined by the distribution span({ g ₁, g ₂}). (See figure 12.9 for a drawing of a foliation.) Is the distribution regular?

For the vector fields g ₁ = [1, 0, 0] ^T and g ₂ = [0, 1, 0] ^T on , sketch the foliation defined by the distribution span({ g ₁, g ₂}). Is the distribution regular? Is it involutive?

For the vector fields g ₁ = [ x ₃,0,0] and g ₂ = [0,1,0] ^T , sketch the foliation defined by the distribution span({ g ₁, g ₂}). Is the distribution regular? Is it involutive?

For the vector fields g ₁ = [1+ x ₃,1 ? x ₂,0] ^T and g ₂ = [0,0,1] ^T on , sketch the foliation defined by the distribution...