Differential form

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, differential forms provide a unified approach to define <a href="/facts/Integrand/V7lyV997">integrands</a> over curves, surfaces, solids, and higher-dimensional <a href="/facts/Manifold/rXPERJtu">manifolds</a>. The modern notion of differential forms was pioneered by <a href="/facts/%25C3%2589lie_Cartan/Uvhkp66S">Élie Cartan</a>. It has many applications, especially in geometry, topology and physics.
For instance, the expression 
 
 
 
 f
 (
 x
 )
 
 d
 x
 
 
 {\displaystyle f(x)\,dx}
 
 is an example of a <a href="/facts/1-form/NQLbWmtD">1-form</a>, and can be <a href="/facts/Integral/4PeM0mJc">integrated</a> over an interval 
 
 
 
 [
 a
 ,
 b
 ]
 
 
 {\displaystyle [a,b]}
 
 contained in the domain of 
 
 
 
 f
 
 
 {\displaystyle f}
 
:

∫
          
            a
          
          
            b
          
        
        f
        (
        x
        )
        
        d
        x
        .
      
    
    {\displaystyle \int _{a}^{b}f(x)\,dx.}

Similarly, the expression 
 
 
 
 f
 (
 x
 ,
 y
 ,
 z
 )
 
 d
 x
 ∧
 d
 y
 +
 g
 (
 x
 ,
 y
 ,
 z
 )
 
 d
 z
 ∧
 d
 x
 +
 h
 (
 x
 ,
 y
 ,
 z
 )
 
 d
 y
 ∧
 d
 z
 
 
 {\displaystyle f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz}
 
 is a 2-form that can be integrated over a <a href="/facts/Surface_(mathematics)/whF08jvy">surface</a> 
 
 
 
 S
 
 
 {\displaystyle S}
 
:

∫
          
            S
          
        
        
          (
          
            f
            (
            x
            ,
            y
            ,
            z
            )
            
            d
            x
            ∧
            d
            y
            +
            g
            (
            x
            ,
            y
            ,
            z
            )
            
            d
            z
            ∧
            d
            x
            +
            h
            (
            x
            ,
            y
            ,
            z
            )
            
            d
            y
            ∧
            d
            z
          
          )
        
        .
      
    
    {\displaystyle \int _{S}\left(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge dx+h(x,y,z)\,dy\wedge dz\right).}

The symbol 
 
 
 
 ∧
 
 
 {\displaystyle \wedge }
 
 denotes the <a href="/facts/Exterior_product/rdN4TvpR">exterior product</a>, sometimes called the wedge product, of two differential forms. Likewise, a 3-form 
 
 
 
 f
 (
 x
 ,
 y
 ,
 z
 )
 
 d
 x
 ∧
 d
 y
 ∧
 d
 z
 
 
 {\displaystyle f(x,y,z)\,dx\wedge dy\wedge dz}
 
 represents a <a href="/facts/Volume_element/SD2ppCLr">volume element</a> that can be integrated over a region of space. In general, a k-form is an object that may be integrated over a k-dimensional manifold, and is <a href="/facts/Homogeneous_polynomial/WUHL7Kct">homogeneous</a> of degree k in the coordinate differentials 
 
 
 
 d
 x
 ,
 d
 y
 ,
 …
 .
 
 
 {\displaystyle dx,dy,\ldots .}

On an n-dimensional manifold, a top-dimensional form (n-form) is called a <a href="/facts/Volume_form/2x37pDLg">volume form</a>.
The differential forms form an <a href="/facts/Alternating_algebra/kDoTSQXn">alternating algebra</a>. This implies that 
 
 
 
 d
 y
 ∧
 d
 x
 =
 −
 d
 x
 ∧
 d
 y
 
 
 {\displaystyle dy\wedge dx=-dx\wedge dy}
 
 and 
 
 
 
 d
 x
 ∧
 d
 x
 =
 0.
 
 
 {\displaystyle dx\wedge dx=0.}
 
 This alternating property reflects the <a href="/facts/Orientation_(mathematics)/lGXs42Nw">orientation</a> of the domain of integration. 
The <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a> is an operation on differential forms that, given a k-form 
 
 
 
 φ
 
 
 {\displaystyle \varphi }
 
, produces a (k+1)-form 
 
 
 
 d
 φ
 .
 
 
 {\displaystyle d\varphi .}
 
 This operation extends the <a href="/facts/Differential_of_a_function/337XJVMC">differential of a function</a> (a function can be considered as a 0-form, and its differential is 
 
 
 
 d
 f
 (
 x
 )
 =
 
 f
 ′
 
 (
 x
 )
 
 d
 x
 
 
 {\displaystyle df(x)=f'(x)\,dx}
 
). This allows expressing the <a href="/facts/Fundamental_theorem_of_calculus/pHeSGtUx">fundamental theorem of calculus</a>, the <a href="/facts/Divergence_theorem/pUGKF1N2">divergence theorem</a>, <a href="/facts/Green%2527s_theorem/WBDk654A">Green's theorem</a>, and <a href="/facts/Stokes%2527_theorem/tNBH6MKK">Stokes' theorem</a> as special cases of a single general result, the <a href="/facts/Generalized_Stokes_theorem/tNBH6MKK">generalized Stokes theorem</a>.
Differential 1-forms are naturally dual to <a href="/facts/Vector_field/ccFNuPPp">vector fields</a> on a <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifold</a>, and the pairing between vector fields and 1-forms is extended to arbitrary differential forms by the <a href="/facts/Interior_product/DjfNykHd">interior product</a>. The algebra of differential forms along with the exterior derivative defined on it is preserved by the <a href="/facts/Pullback_(differential_geometry)/EzMDBYfK">pullback</a> under smooth functions between two manifolds. This feature allows geometrically invariant information to be moved from one space to another via the pullback, provided that the information is expressed in terms of differential forms. As an example, the <a href="/facts/Change_of_variables_formula/NvXfTTaf">change of variables formula</a> for integration becomes a simple statement that an integral is preserved under pullback.

Differential form open-in-new

Differential form