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Examples of path functions include work, heat and arc length. In contrast to path functions, state functions are independent of the path taken. Thermodynamic state variables are point functions, differing from path functions. For a given state, considered as a point, there is a definite value for each state variable and state function.
The difference between initial and final states of the system's internal energy does not account for the extent of the energy interactions transpired. Therefore, internal energy is a state function (i.e. exact differential), while heat and work are path functions (i.e. inexact differentials) because integration must account for the path taken.
The path can be specified by noting the values of the state parameters as the system traces out the path, whether as a function of time or a function of some other external variable. For example, having the pressure P ( t ) and volume V ( t ) as functions of time from time t 0 to t 1 will specify a path in two-dimensional state space.
They are uniquely determined by the thermodynamic state as it has been identified by the original state variables. There are many such state functions. Examples are internal energy, enthalpy, Helmholtz free energy, Gibbs free energy, thermodynamic temperature, and entropy. For a given body, of a given chemical constitution, when its ...
is called a sample function, a realization, or, particularly when is interpreted as time, a sample path of the stochastic process {(,):}. [50] This means that for a fixed ω ∈ Ω {\displaystyle \omega \in \Omega } , there exists a sample function that maps the index set T {\displaystyle T} to the state space S {\displaystyle S} . [ 28 ]
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral.
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
In mathematics, a path in a topological space is a continuous function from a closed interval into . Paths play an important role in the fields of topology and mathematical analysis . For example, a topological space for which there exists a path connecting any two points is said to be path-connected .