4.1. $(x d x+y d y) \wedge(-y d x+x d y)$
Answer
\(\boxed{0}\) is the wedge product of the two differential forms \((x dx + y dy)\) and \((-y dx + x dy)\).
Step 1 ：Given two differential forms: \((x dx + y dy)\) and \((-y dx + x dy)\). We need to calculate their wedge product.
Step 2 ：The wedge product of two differential forms is calculated by taking the product of the coefficients and the wedge product of the differentials.
Step 3 ：The wedge product of differentials is calculated by treating them as vectors and taking their cross product.
Step 4 ：In this case, the two forms are orthogonal to each other, meaning they are at right angles in the vector space.
Step 5 ：The wedge product of two orthogonal vectors is always zero.
Step 6 ：\(\boxed{0}\) is the wedge product of the two differential forms \((x dx + y dy)\) and \((-y dx + x dy)\).
