Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function). Because the derivative is a rate of change, such an equation states how a function changes but does not specify the function itself. Given sufficient initial conditions, however, such as a specific function value, the function can be found by various methods, most based on integration.
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In those hierarchically organised churches of Western Christianity which have an ecclesiastical law system, an ordinary is an officer of the church who by reason of office has ordinary power to execute the church's laws. The term comes from the Latin word ordinarius. In Eastern Christianity, a corresponding officer is called a hierarch, which comes from the Greek word ἱεράρχης meaning "priestly ruler".
The law vesting ordinary power could either be ecclesiastical law, i.e. the positive enactments that the church has established for itself, or divine law, i.e. the laws which the church believes were given to it by God. As an example of divinely instituted ordinaries, Roman Catholics believe that when Jesus established the Church he in turn established the episcopate and the Primacy of Simon Peter and endowed the offices with power to rule the Church. Thus, in the Roman Catholic Church, the office of successor of Simon Peter and the office of diocesan bishop possess their ordinary power even in the absence of positive enactments from the Church.
Many officers possess ordinary power but, due to their lack of ordinary executive power, are not called ordinaries. The best example of this phenomenon is the office of judicial vicar, a.k.a. officialis. The judicial vicar only has authority through his office to exercise the diocesan bishop's power to judge cases. Though the vicar has vicarious ordinary judicial power, he is not an ordinary because he lacks ordinary executive power. A vicar general, however, has authority through his office to exercise the diocesan bishop's executive power. He is therefore an ordinary because of this vicarious ordinary executive power.
Local ordinaries are ordinaries over particular churches. The following officers are local ordinaries:
Other officers are also ordinaries (Latin Church) or hierarchs (Eastern Churches), but not local ordinaries (Latin Church) or local hierarchs (Eastern Churches):
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