I was convinced initially by the definition which differential forms are defined through general definition of tensor field, i.e. $$\omega:M\to T^r_s(M)$$ where differential 1-form is just the particular case that $T^r_s(M)=T^0_1=T^*(M)$.
However, when reading Complex Variables An Introduction by Carlos A. Berenstein, Roger Gay, it defines differential 1-form as below
Could someone instruct me a bit about how to see the equivalence of these two definitions? (or are they equivalent?)
And also, the later chapters of this book is apparently too difficult for me, is there any alternative books that have similar treating (complex analysis through differential forms)?