On ℝ³ with coordinates (x, y, z), the exterior derivative takes: 0-forms (functions) to 1-forms via df = (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz. This corresponds to which classical vector calculus operation?
ADivergence
BCurl
CGradient
DLaplacian
The exterior derivative of a 0-form (function) on ℝ³ gives a 1-form whose components are the partial derivatives — this is exactly the gradient. Continuing the correspondence: d on 1-forms corresponds to the curl (d of a 1-form gives a 2-form whose components are the curl), and d on 2-forms corresponds to the divergence (d of a 2-form gives a 3-form whose coefficient is the divergence). The identity d² = 0 unifies curl(grad f) = 0 and div(curl F) = 0.
Question 2 True / False
A differential form ω is called closed if dω = 0, and exact if ω = dα for some form α. Every exact form is closed. Is every closed form exact?
TTrue
FFalse
Answer: False
Not in general. The famous counterexample is the 1-form ω = (-y dx + x dy)/(x² + y²) on ℝ² {0}. It is closed (dω = 0) but not exact — its integral around the unit circle is 2π, which would be zero if ω = df for some function f. The obstruction to closed forms being exact is topological: the de Rham cohomology group Hᵏ(M) = {closed k-forms}/{exact k-forms} measures this failure. On contractible domains, the Poincaré lemma guarantees every closed form is exact.
Question 3 Short Answer
Why does d² = 0 hold? What is the essential reason?
Think about your answer, then reveal below.
Model answer: The identity d² = 0 follows from the equality of mixed partial derivatives (∂²f/∂xⁱ∂xʲ = ∂²f/∂xʲ∂xⁱ) combined with the antisymmetry of the wedge product (dxⁱ ∧ dxʲ = -dxʲ ∧ dxⁱ). When you apply d twice, symmetric second derivatives get wedged with antisymmetric basis forms, and every term cancels with its partner.
In coordinates: d(df) = d(∂f/∂xⁱ dxⁱ) = (∂²f/∂xʲ∂xⁱ) dxʲ ∧ dxⁱ. Since ∂²f/∂xʲ∂xⁱ is symmetric in i,j while dxʲ ∧ dxⁱ is antisymmetric, every pair of terms cancels. This extends to k-forms by the Leibniz rule. The identity d² = 0 is the manifold version of 'curl of gradient is zero' and 'divergence of curl is zero' — both of which are consequences of the same symmetry-antisymmetry cancellation.
Question 4 True / False
The exterior derivative d is uniquely characterized by four properties: (1) d takes k-forms to (k+1)-forms, (2) on 0-forms, df(X) = X(f), (3) d² = 0, and (4) d(α ∧ β) = dα ∧ β + (-1)^deg(α) α ∧ dβ.
TTrue
FFalse
Answer: True
These four properties (degree-raising, agreement with the differential on functions, nilpotency, and the graded Leibniz rule) uniquely determine d. This is important because it means d is coordinate-independent — any coordinate formula satisfying these axioms must give the same operator. The graded Leibniz rule (the sign (-1)^k accounts for the degree of α) ensures compatibility with the wedge product's antisymmetry.