What does the differential $l$-form equation mean?

Question

I have this equation which is from here (equation 12.63) of the book "Differential Geometry, Gauge Theories, and Gravity". Can somebody explain what it means? thanks!

the equation is:

Answer 1

This is just the definition of the exterior derivative, which generalizes the gradient operation from old-fashioned calculus on $\mathbb{R}^n$.