ASSIGNMENT: THE WEDGE PRODUCT AND THE PRODUCT OPERATIONS FROM VECTOR CALCULUS

The wedge product might seem weird to you at first. One way to become more comfortable with it is to see how it is connected to concepts that you are already familiar with. In this assignment, you will learn about how the wedge product is related to determinants, cross products, and other products from vector calculus.

1. The 2 × 2 determinant

Exercise 1 - Let α and β be the 1-forms on R² that correspond to vectors a and b. Show that α ∧ β = (a b)dx ∧ dy.

2. The 2 × 2 determinant and the triple scalar product

Exercise 2 - Let α, β, and γ be 1-forms (with constant coefficients) on R³, and let A be the matrix whose rows are the corresponding vectors. Show that α ∧ β ∧ γ = |A|dx ∧ dy ∧ dz.

In 3 dimensions, there is another formula for the determinant, using something called the triple scalar product. (You probably encountered it in Calc III, but it's likely that you only saw it briefly and then never again, so don't feel bad if you don't remember.)

Exercise 3 - Let a, b, and c be vectors in R³, and let A be the matrix whose rows are a, b, and c. Show that det A = T(a, b, c).

3. The dot and cross products

Exercise 4 - Let α and β be the 1-forms on R³ that correspond to vectors a and b. Show that α ∧ β corresponds to a × b.

Exercise 5 - Let a and b be vectors in R³. Let α be the 1-form corresponding to a, and let β be the 2-form corresponding to b. Show that α ∧ β = (a · b)dx ∧ dy ∧ dz.

What about the other possibilities?

Exercise 6 - Let a and b be vectors in R³.

(a) Let α be the 2-form corresponding to a, and let β be the 1-form corresponding to b. What does α ∧ β calculate?

(b) Let α be the 2-form corresponding to a, and let β be the 2-form corresponding to b. What does α ∧ β calculate?

Exercise 7 - Use the results of Exercises 3, 4, and 5 to give a "coordinate-free" proof of Exercise 2. (Coordinate-free means that you avoid calculations where forms are written out in terms of dx, dy, and dz.)

4. Determinants in higher dimensions

Exercise 8 - Let α₁, . . . α_n be 1-forms (with constant coefficients) on Rⁿ, and let A be the matrix whose rows are the corresponding vectors. Show that α₁ ∧ α₂ ∧ · · · ∧ α_n = |A|dx₁ ∧ dx₂ ∧ · · · ∧ dx_n.

Remark: For this exercise, you'll probably want to use a double-index, i.e.

α₁ = a₁₁dx₁ + a₁₂dx₂ + · · · + a_1ndx_n,

α2 = a₂₁dx₁ + a₂₂dx₂ + · · · + a_2ndx_n,

etc.

Attachment:- Assignment File.rar