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A total differential equation is a differential equation expressed in terms of total derivatives. Since the exterior derivative is coordinate-free, in a sense that can be given a technical meaning, such equations are intrinsic and geometric.
A number of properties of the differential follow in a straightforward manner from the corresponding properties of the derivative, partial derivative, and total derivative. These include: [ 11 ] Linearity : For constants a and b and differentiable functions f and g , d ( a f + b g ) = a d f + b d g . {\displaystyle d(af+bg)=a\,df+b\,dg.}
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.
Another two-point formula is to compute the slope of a nearby secant line through the points (x − h, f(x − h)) and (x + h, f(x + h)). The slope of this line is (+) (). This formula is known as the symmetric difference quotient.
Given a simply connected and open subset D of and two functions I and J which are continuous on D, an implicit first-order ordinary differential equation of the form (,) + (,) =,is called an exact differential equation if there exists a continuously differentiable function F, called the potential function, [1] [2] so that
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. [1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
An ordinary differential equation is a differential equation that relates functions of one variable to their derivatives with respect to that variable. A partial differential equation is a differential equation that relates functions of more than one variable to their partial derivatives. Differential equations arise naturally in the physical ...
These equations for solution of a first-order partial differential equation are identical to the Euler–Lagrange equations if we make the identification = ˙ ˙. We conclude that the function ψ {\displaystyle \psi } is the value of the minimizing integral A {\displaystyle A} as a function of the upper end point.
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