Question 1
Let φ and ψ be two 1-forms on R
3 say φ= a
1dx
1 + a
2dx
2, ψ= b
1dx
1 + b
2dx
2. Show that φ∧ψ = 0 if and only if φ = c ψ for some constant real number c.
Question 2
Let φ= a
1dx
1 + a
2dx
2 + a
3 dx
3 ∈Λ
1(R
3) and ψ= b
1dx
2∧dx
3 + b
2dx
3∧dx
1+ b
3 dx
1∧dx
2 ∈Λ
2(R
3). If φ and ψ are identified with the vectors
u = a
1 i+ a
2 j+ a
3 k and
v = b
1 i+ b
2 j+ b
3 k, respectively, and if
then c is equal to
◦
u × v◦
◦
u + v◦
u ∙ v◦
+
