In Exercises 4.1-4.5, calculate the indicated product. 4.1. $(x d x+y d y) \wedge(-y d x+x d y)$

Step 1 ：This is a problem of vector calculus, specifically dealing with differential forms and the wedge product. The wedge product is a binary operation on differential forms which is anticommutative and associative. It can be thought of as a generalization of the cross product in three dimensions.

Step 2 ：The wedge product of two differential forms is calculated by distributing the product over the terms and then applying the antisymmetry property.

Step 3 ：In this case, we have two differential forms: \((x dx + y dy)\) and \((-y dx + x dy)\). We need to calculate their wedge product.

Step 4 ：The wedge product of the two differential forms is zero. This is because the two forms are exact differentials of the same function, and the wedge product of an exact differential with itself is always zero.

Step 5 ：Final Answer: The wedge product of \((x dx + y dy)\) and \((-y dx + x dy)\) is \(\boxed{0}\).