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Differential forms can be multiplied together using the exterior product, and for any differential k-form α, there is a differential (k + 1)-form dα called the exterior derivative of α. Differential forms, the exterior product and the exterior derivative are independent of a choice of coordinates.
In mathematics, especially vector calculus and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0), and an exact form is a differential form, α, that is the exterior derivative of another differential form β. Thus, an exact form is in the image of d, and a closed form is in the kernel of d.
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That is, df is the unique 1-form such that for every smooth vector field X, df (X) = d X f , where d X f is the directional derivative of f in the direction of X. The exterior product of differential forms (denoted with the same symbol ∧) is defined as their pointwise exterior product.
The exterior derivative is a notion of differentiation of differential forms which generalizes the differential of a function (which is a differential 1-form). Pullback is, in particular, a geometric name for the chain rule for composing a map between manifolds with a differential form on the target manifold.
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The wedge product of complex differential forms is defined in the same way as with real forms. Let p and q be a pair of non-negative integers ≤ n. The space Ω p,q of (p, q)-forms is defined by taking linear combinations of the wedge products of p elements from Ω 1,0 and q elements from Ω 0,1. Symbolically,
Let M be a smooth manifold and E → M be a smooth vector bundle over M.We denote the space of smooth sections of a bundle E by Γ(E).An E-valued differential form of degree p is a smooth section of the tensor product bundle of E with Λ p (T ∗ M), the p-th exterior power of the cotangent bundle of M.