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Substitution, written M[x := N], is the process of replacing all free occurrences of the variable x in the expression M with expression N. Substitution on terms of the lambda calculus is defined by recursion on the structure of terms, as follows (note: x and y are only variables while M and N are any lambda expression): x[x := N] = N
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution, [6] and also known by variant names such as half-tangent substitution or half-angle substitution.
In geometry, an envelope of a planar family of curves is a curve that is tangent to each member of the family at some point, and these points of tangency together form the whole envelope. Classically, a point on the envelope can be thought of as the intersection of two " infinitesimally adjacent" curves, meaning the limit of intersections of ...
One most important example is the "substitution lemma", which with the notation of λx becomes (M x:=N ) y:=P = (M y:=P ) x:=(N y:=P ) (where x≠y and x not free in P) A surprising counterexample, due to Melliès, [ 5 ] shows that the way this rule is encoded in the original calculus of explicit substitutions is not strongly normalizing .
The substitution rule states that for any φ and any term t, one can conclude φ[t/x] from φ provided that no free variable of t becomes bound during the substitution process. (If some free variable of t becomes bound, then to substitute t for x it is first necessary to change the bound variables of φ to differ from the free variables of t .)
Note that the bounds of integration may as well be π and − π, as in the previous example, or any other pair of endpoints 2 π apart. The trick is to use the substitution z = e it where dz = ie it dt and hence =. This substitution maps the interval [0, 2π] to the unit circle.
In calculus, integration by substitution, also known as u-substitution, reverse chain rule or change of variables, [1] is a method for evaluating integrals and antiderivatives. It is the counterpart to the chain rule for differentiation , and can loosely be thought of as using the chain rule "backwards."
In mathematics, a trigonometric substitution replaces a trigonometric function for another expression. In calculus, trigonometric substitutions are a technique for evaluating integrals. In this case, an expression involving a radical function is replaced with a trigonometric one. Trigonometric identities may help simplify the answer.