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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.
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 ...
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, 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.
Total derivative, total differential and Jacobian matrix Main article: Total derivative When f {\displaystyle f} is a function from an open subset of R n {\displaystyle \mathbb {R} ^{n}} to R m {\displaystyle \mathbb {R} ^{m}} , then the directional derivative of f {\displaystyle f} in a chosen direction is the best linear approximation ...
where is the total infinitesimal change in for an infinitesimal displacement , and is seen to be maximal when is in the direction of the gradient . The nabla symbol ∇ {\displaystyle \nabla } , written as an upside-down triangle and pronounced "del", denotes the vector differential operator .
In complex analysis, complex-differentiability is defined using the same definition as single-variable real functions. This is allowed by the possibility of dividing complex numbers . So, a function f : C → C {\textstyle f:\mathbb {C} \to \mathbb {C} } is said to be differentiable at x = a {\textstyle x=a} when
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