the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first...

field of differential geometry, the exterior covariant derivative is an extension of the notion of exterior derivative to the setting of a differentiable...

property reflects the orientation of the domain of integration. The exterior derivative is an operation on differential forms that, given a k-form φ {\displaystyle...

Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in...

In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments...

-1 derivation on the exterior algebra defined by contraction with a vector field), the exterior derivative and the Lie derivative form a Lie superalgebra...

}F_{m}} where Rk are the local basis vectors. Equivalently, using the exterior derivative, the curl can be expressed as: ∇ × F = ( ⋆ ( d F ♭ ) ) ♯ {\displaystyle...

manifold Ω {\displaystyle \Omega } is equal to the integral of its exterior derivative d ω {\displaystyle d\omega } over the whole of Ω {\displaystyle \Omega...

The derivative is a fundamental tool of calculus that quantifies the sensitivity of change of a function's output with respect to its input. The derivative...

Unlike the vector derivative, neither the interior derivative operator nor the exterior derivative operator is invertible. The derivative with respect to...

the exterior derivative is usually easier than working with the vector field and divergence, because unlike the divergence, the exterior derivative commutes...

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held...

the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing...

the exterior derivative gives the exterior algebra of differential forms on a manifold the structure of a differential graded algebra. The exterior derivative...

the exterior derivative d , {\displaystyle d,} which has the property d ∘ d = 0. {\displaystyle d\circ d=0.} The interior product relates the exterior derivative...

product rule. The Lie derivative is another derivative that is covariant under basis transformations. Like the exterior derivative, it does not depend on...

total differential or exterior derivative of f {\displaystyle f} and is an example of a differential 1-form. Much as the derivative of a function of a single...

the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case...

vector to M at p, then the directional derivative of f along v, denoted variously as df(v) (see Exterior derivative), ∇ v f ( p ) {\displaystyle \nabla _{\mathbf...

electromagnetic tensor, conventionally labelled F, is defined as the exterior derivative of the electromagnetic four-potential, A, a differential 1-form:...

an exterior covariant derivative d ∇ : Ω r ( E ) → Ω r + 1 ( E ) . {\displaystyle d_{\nabla }:\Omega ^{r}(E)\to \Omega ^{r+1}(E).} This exterior covariant...

theorem—generalize to a theorem (also called Stokes' theorem) relating the exterior derivative and integration over submanifolds. Differentiable functions between...

complex of differential forms on some smooth manifold M, with the exterior derivative as the differential: 0 → Ω 0 ( M )   → d   Ω 1 ( M )   → d   Ω 2...

computing derivatives of functions Exterior calculus identities Exterior derivative – Operation on differential forms List of limits Table of derivatives – Rules...

basis, the connection form transforms in a manner that involves the exterior derivative of the transition functions, in much the same way as the Christoffel...

space V there is a natural exterior derivative on the space of V-valued forms. This is just the ordinary exterior derivative acting component-wise relative...

covariant derivative. Alternatively, the operator can be generalized to operate on differential forms using the divergence and exterior derivative. The resulting...

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...

dxω is the exterior derivative of ω with respect to the space variables only and ω ˙ {\displaystyle {\dot {\omega }}} is the time derivative of ω. If we...

boundary ∂M of an n-dimensional manifold M to the integral of dω (the exterior derivative of ω, and a differential n-form on M) over M itself: ∫ M d ω = ∫...