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Differential forms are the objects you integrate on manifolds. A differential k-form assigns to each point a multilinear, alternating function on k tangent vectors. 0-forms are smooth functions, 1-forms are dual to vector fields (they eat a vector and return a number), and higher forms generalize the integrands of line, surface, and volume integrals. The exterior algebra provides a coordinate-free framework for multivariable calculus on manifolds.
You know that a tangent vector at p is a linear map from functions to numbers. Dually, a covector (or 1-form at p) is a linear map from tangent vectors to numbers: ω_p : TpM → ℝ. The space of all covectors at p is the cotangent space T*pM, which is the dual vector space to TpM. If (x¹, ..., xⁿ) are local coordinates, the differentials dx¹, ..., dxⁿ form a basis for T*pM, dual to the basis ∂/∂x¹, ..., ∂/∂xⁿ of TpM. A smooth 1-form is a smooth section of the cotangent bundle — a smooth choice of covector at each point.
The most fundamental 1-form is the differential of a smooth function f, written df. It acts on a tangent vector v by df(v) = v(f). In coordinates, df = (∂f/∂xⁱ)dxⁱ. This is the coordinate-free version of the gradient — but unlike the gradient (which is a vector field), df requires no metric. The distinction between df (a 1-form) and ∇f (a vector field) is central to differential geometry: they carry the same information, but converting between them requires a Riemannian metric via the "musical isomorphism" ♯ and ♭.
Higher-degree forms are built from the wedge product ∧, which is the antisymmetrized tensor product. A k-form at p is an alternating multilinear map ω_p : (TpM)ᵏ → ℝ. "Alternating" means swapping two inputs flips the sign: ω(v, w) = -ω(w, v). The wedge product of a k-form and an l-form is a (k+l)-form, satisfying α ∧ β = (-1)kl β ∧ α. In coordinates on an n-manifold, every k-form is a sum of terms like f_{i₁...iₖ} dxⁱ¹ ∧ ... ∧ dxⁱᵏ with i₁ < ... < iₖ. Since alternating forms on an n-dimensional space vanish when k > n, there are no forms of degree greater than n.
The reason differential forms are central to geometry is integration. A k-form can be integrated over a k-dimensional oriented submanifold, and the result is independent of the coordinate system used — the alternating, multilinear structure of forms is precisely what makes the change-of-variables formula work automatically. In ℝ³, 1-forms correspond to line integrands, 2-forms to surface integrands, and 3-forms to volume integrands. The exterior derivative (upcoming) and Stokes' theorem complete the package, unifying the classical integral theorems of vector calculus into a single elegant framework.