Homework
Q1. What is the configuration space of all pentagons in the plane with fixed side lengths a, b, c, d, e? You can assume the side lengths are sufficiently irregular to avoid singularities. Also, you can assume that one of the edges is fixed, as was done in the examples involving quadrilaterals.
Q2. Consider the forms α = xdx - ydy, β = zdx Λ dy + xdy Λ dz, and γ = zdy on R3. Calculate the following:
(a) α Λ β
(b) α Λ γ
Q3. Let α, β, and γ be the same as in the previous problem. For each of the following expressions, give a one-sentence explanation of why the result is 0.
(a) α Λ α
(b) β Λ β
(c) α Λ γ + γ Λ α
(d) α Λ β Λ γ
Q4. Consider R4, where the coordinates are labeled (x1, y1, x2, y2). Let ω = dx1 Λ dy1 + dx2 Λ dy2. Compute ω Λ ω.
Q5. Sometimes (for example in electromagnetism) it is convenient to work with 4-dimensional space-time, where there are three space variables x, y, z and a time variable t. Since the time variable plays a special role, it is frequently given special treatment. Specifically, it is common to factor out dt whenever it appears in differential forms. For example, if a = x2tdx Λ dy + xyzdz Λ dt + yztdx Λ dt, then we can write
α = x2tdx Λ dy + xyzdz Λ dt + yztdx Λ dt
= x2tdx Λ dy + (xyzdz + yztdx) Λ dt.
More generally, any differential form α can be uniquely written as α = αs + αt Λ dt,
where αs and αt are forms that are "spatial" (meaning that they don't have dt in them; but they can still depend on t). Conversely, given the spatial forms a, and at, you can use the above formula to reconstruct α.
This gives us a correspondence between differential forms α on space time and pairs of spatial forms (αs, αt).
(a) If α is a k-form, then what are the degrees of αs and αt?
(b) Find a formula for the wedge product in terms of pairs of spatial forms. In other words, if (αs, αt) and (βs, βt) are pairs corresponding to a k-form α and l-form β, then find a formula (in terms of αs, αt, βs, βt) for the pair of spatial forms that corresponds to α Λ β.
(c) Suppose that ω is a 2-form on space time, corresponding to the pair (ωs, ωt). Show that ω Λ ω corresponds to the pair (0, 2ωs Λ ωt).