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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.
In mathematics, an exact differential equation or total differential equation is a certain kind of ordinary differential equation which is widely used in physics and engineering. Definition [ edit ]
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.
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.}
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.
The term differential is used nonrigorously in calculus to refer to an infinitesimal ("infinitely small") change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an ...
Differential equations arise naturally in the physical sciences, in mathematical modelling, and within mathematics itself. For example, Newton's second law, which describes the relationship between acceleration and force, can be stated as the ordinary differential equation =.
This notation is popular in physics and mathematical physics. It also appears in areas of mathematics connected with physics such as differential equations. When taking the derivative of a dependent variable y = f(x), an alternative notation exists: [16]