Detailed solution to Problem Set 43 Exercise 5

Question:
Find the exterior differential (aka exterior derivative) of the following exterior differential form: 2x dx ∧ dy + 3y dx ∧ dz + 4z dy ∧ dz.
Solution:
d ∧ (2x dx ∧ dy + 3y dx ∧ dz + 4z dy ∧ dz) = d(2x) ∧ dx ∧ dy + d(3y) ∧ dx ∧ dz + d(4z) ∧ dy ∧ dz = 2 dx ∧ dx ∧ dy + 3 dy ∧ dx ∧ dz + 4 dz ∧ dy ∧ dz = 0 − 3 dx ∧ dy ∧ dz + 0 = −3 dx ∧ dy ∧ dz.

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