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Concept
Exact differential
Résumé
In multivariate calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is equal to the general differential for some differentiable function in an orthogonal coordinate system (hence is a multivariable function whose variables are independent, as they are always expected to be when treated in multivariable calculus).
An exact differential is sometimes also called a total differential, or a full differential, or, in the study of differential geometry, it is termed an exact form.
The integral of an exact differential over any integral path is path-independent, and this fact is used to identify state functions in thermodynamics.
Even if we work in three dimensions here, the definitions of exact differentials for other dimensions are structurally similar to the three dimensional definition. In three dimensions, a form of the type
is called a differential form. This form is called exact on an open domain in space if there exists some differentiable scalar function defined on such that
throughout , where are orthogonal coordinates (e.g., Cartesian, cylindrical, or spherical coordinates). In other words, in some open domain of a space, a differential form is an exact differential if it is equal to the general differential of a differentiable function in an orthogonal coordinate system.
Note: In this mathematical expression, the subscripts outside the parenthesis indicate which variables are being held constant during differentiation. Due to the definition of the partial derivative, these subscripts are not required, but they are explicitly shown here as reminders.
The exact differential for a differentiable scalar function defined in an open domain is equal to , where is the gradient of , represents the scalar product, and is the general differential displacement vector, if an orthogonal coordinate system is used. If is of differentiability class (continuously differentiable), then is a conservative vector field for the corresponding potential by the definition.
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