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Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations. A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is a straight line between the points. However, if the curve is constrained to ...
A coefficient is usually a constant quantity, but the differential coefficient of f is a constant function only if f is a linear function. When f is not linear, its differential coefficient is a function, call it f ′, derived by the differentiation of f, hence, the modern term, derivative. The older usage is now rarely seen.
An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments). [ 56 ] : 204–206 Thus, an implicit function for y {\displaystyle y} in the context of the unit circle is defined implicitly by x 2 + f ( x ) 2 − 1 = 0 {\displaystyle x^{2}+f ...
In propositional calculus a literal is simply a propositional variable or its negation.. In predicate calculus a literal is an atomic formula or its negation, where an atomic formula is a predicate symbol applied to some terms, (, …,) with the terms recursively defined starting from constant symbols, variable symbols, and function symbols.
In the context of functions, the term variable refers commonly to the arguments of the functions. This is typically the case in sentences like "function of a real variable", "x is the variable of the function f: x ↦ f(x)", "f is a function of the variable x" (meaning that the argument of the function is referred to by the variable x).
For a differential equation parameterized on time, the variable's evolution is stable if and only if the real part of each root is negative. For difference equations, there is stability if and only if the modulus of each root is less than 1. For both types of equation, persistent fluctuations occur if there is at least one pair of complex roots.
Heuristically, is the change in , so we 'formally' have =, and then this is similar in form to the total differential of a function (,, …,), = = , where ,, …, are independent variables. Comparing the last two equations, the functional derivative / has a role similar to that of the partial derivative /, where the variable of integration is ...
Such equations give rise to the terminology found in some texts wherein the derivative is referred to as the "differential coefficient" (i.e., the coefficient of dx). Some authors and journals set the differential symbol d in roman type instead of italic: dx. The ISO/IEC 80000 scientific style guide recommends this style.