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 …

Definition. The exterior derivative of a differential form of degree k (also differential k-form, or just k-form for brevity here) is a differential form of degree k + 1.. If f is a smooth function (a 0-form), then the exterior derivative of f is the differential of f .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 ...

The exterior derivative commutes with pullback along smooth mappings between manifolds, and it is therefore a natural differential operator. The exterior algebra of differential forms, equipped with the exterior derivative, is a cochain complex whose cohomology is called the de Rham cohomology of the underlying manifold and plays a vital role ...

In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable.The differential dy is defined by = ′ (), where ′ is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx).The notation is such that the equation

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.

In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix.It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism.The determinant of a product of …

This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and is particularly useful in the differentiation of integral transforms.An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable.

In the differential form formulation on arbitrary space times, F = 1 / 2 F αβ dx α ∧ dx β is the electromagnetic tensor considered as a 2-form, A = A α dx α is the potential 1-form, = is the current 3-form, d is the exterior derivative, and is the Hodge star on forms defined (up to its orientation, i.e. its sign) by the Lorentzian ...

Elementary rules of differentiation. Unless otherwise stated, all functions are functions of real numbers that return real values; although more generally, the formulae below apply wherever they are well defined — including the case of complex numbers ().. Constant Term Rule. For any value of , where , for any value of , () =.. Proof. Let .Now, we will prove, from first principles, what the ...