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Derivation

Derivation is a term with multiple meanings across various fields, including linguistics, law, music, and science and mathematics (see sections below for details). In linguistics, it refers to the morphological process of creating new words from existing ones, typically by adding affixes such as prefixes or suffixes to a base or root, which often alters the word's meaning, grammatical category, or both.^[1] This mechanism is a primary means of vocabulary expansion in many languages, enabling the formation of related but distinct lexemes, such as transforming the adjective happy into the noun happiness through the addition of the suffix -ness. Derivational morphology differs from inflectional morphology in that it produces novel words that can function independently as new lexical items, rather than merely modifying a word for grammatical purposes like tense or number.^[2] Common derivational processes include affixation (e.g., teach to teacher via the suffix -er, denoting an agent).^[3] Other processes, such as zero-derivation or conversion (e.g., run as both verb and noun without affixation) and reduplication in some languages, are also used. These operations are productive to varying degrees across languages; for instance, English favors suffixation for deriving nouns from verbs (e.g., decide to decision), while languages like Turkish exhibit extensive agglutinative derivation through vowel harmony and suffix chains. The study of derivation reveals insights into language structure and evolution, as derivational patterns often reflect semantic relationships, such as nominalization (verb to noun) or denominalization (noun to adjective), and contribute to the lexicon's adaptability.^[4] Derivation modifies a single base, in contrast to compounding, which combines two or more roots, though the two processes can interact in complex word formation. In generative linguistics, derivation is analyzed through rules that map underlying forms to surface realizations, highlighting productivity constraints based on phonological, syntactic, and semantic factors.^[5]

Language

Morphological Derivation

Morphological derivation refers to the linguistic process by which new words, or lexemes, are created from existing roots or bases through the attachment of affixes, including prefixes, suffixes, and infixes, often resulting in changes to the word's meaning or grammatical category.^[6] This mechanism expands a language's vocabulary by producing derived forms that can shift categories, such as converting a verb into a noun or an adjective into an adverb, while preserving a semantic connection to the base form.^[7] For instance, in English, the addition of affixes allows for the creation of words like darkness (suffix -ness deriving a noun from the adjective dark) or refusal (suffix -al deriving a noun from the verb refuse), illustrating how derivation modifies lexical items to express nuanced concepts. The foundations of morphological derivation as a distinct category trace back to structural linguistics in the early 20th century, with significant contributions from Leonard Bloomfield, who in his 1933 seminal work Language clearly differentiated derivation from inflection by emphasizing derivation's role in forming new words with altered lexical properties rather than merely marking grammatical relations.^[8] Bloomfield's analysis, rooted in descriptive linguistics, highlighted how derivational processes operate on morphemes to build complex words, influencing subsequent morphological theories that view derivation as a productive system for lexical innovation. This distinction became central to understanding word formation, as derivation typically involves bound morphemes that carry lexical meaning, unlike inflectional endings that indicate syntax or agreement without creating novel lexemes.^[9] A key feature of morphological derivation is its reliance on a single root or base combined with one or more affixes, setting it apart from compounding, which merges two or more independent roots or stems to form a new word. For example, English unhappiness derives from the root happy via the prefix un- and suffix -ness, maintaining a unitary base structure, whereas a compound like raincoat integrates the roots rain and coat without affixation.^[1] This contrast underscores derivation's focus on affix-driven modification, enabling languages to generate extensive derived vocabularies efficiently while avoiding the multi-root synthesis characteristic of compounding.^[10]

Derivational Processes

Derivational processes encompass a range of mechanisms beyond simple affixation, enabling the formation of new words through non-concatenative means such as zero-derivation, subtractive derivation, and reduplication. Zero-derivation, also known as conversion, involves shifting a word's grammatical category without any overt morphological change, relying instead on syntactic context to signal the new function. For instance, the English word "run" functions as both a verb (to run) and a noun (a run) through this process, allowing speakers to repurpose existing lexical items efficiently. Subtractive derivation, a rarer process, creates new forms by removing segments from a base word, often in gender or number marking. In French, subtractive morphology is evident in the gender agreement of adjectives, where the masculine form is derived by deleting the final -e from the feminine, as in "gros" (masculine 'large') from "grosse" (feminine). This mechanism contrasts with more common additive processes and highlights morphological economy in certain languages. Reduplication involves partial or full repetition of a base to derive new meanings, frequently marking aspect, plurality, or intensity. In Tagalog, the verb "takbo" (run) becomes "tatakbo" (will run) through initial syllable reduplication, signaling future tense and demonstrating how this process integrates phonological and semantic shifts in Austronesian languages.^[11] The productivity of derivational affixes varies significantly, determining their potential to generate novel words and fill expressive needs in a language. Productivity is quantified through measures like the ratio of hapax legomena (unique occurrences) to total tokens, reflecting an affix's openness to neologisms. In English, the suffix "-ness," which forms abstract nouns from adjectives (e.g., kind → kindness), exhibits high productivity, with 2,466 distinct types and 943 hapax legomena in the British National Corpus's written subcorpus.^[12] Similarly, the adverb-forming suffix "-ly" (e.g., quick → quickly) is among the most productive, appearing in thousands of forms and enabling extensive adverbial derivation from adjectives, though its exact count varies by corpus due to the language's analytic tendencies.^[13] Factors influencing productivity include phonological compatibility with bases and semantic transparency, with highly productive affixes like these showing low type-token ratios and frequent innovation. Cross-linguistically, derivational processes reveal unique constraints and innovations. In German, nominal derivation from verbs often occurs via zero-derivation combined with obligatory capitalization, transforming the infinitive "gehen" (to go/walk) into the noun "das Gehen" (the act of going/walking), where the capital signals nominal status without additional affixes. Constraints such as blocking further regulate derivation by preempting potential forms when a synonymous word already exists. For example, the potential derivation "*highth" from "high" using the abstract noun suffix "-th" (as in width) is blocked by the established form "height," preventing redundancy and maintaining lexical economy in English morphology. These processes play a key role in language evolution by addressing lexical gaps—absences in vocabulary for emerging concepts—and facilitating adaptation over time. Derivation allows speakers to coin terms like "arrival" from "arrive" to denote the resulting state, filling semantic voids without borrowing and enabling the lexicon to expand organically as societal needs change.^[14] Through such mechanisms, languages evolve dynamically, with productive derivations contributing to lexical renewal and preventing stagnation.

Law

Derivative Works in Copyright

In copyright law, a derivative work is defined as a work based upon one or more preexisting works, such as a translation, musical arrangement, dramatization, fictionalization, motion picture version, sound recording, art reproduction, abridgment, condensation, or any other form in which a work may be recast, transformed, or adapted.^[15] This definition, enshrined in the U.S. Copyright Act under 17 U.S.C. § 101, emphasizes that the new work must incorporate substantial elements of the original while adding original authorship through modifications that represent a new creative contribution.^[16] The creator of a derivative work obtains copyright protection only for the new material added, not the underlying preexisting work, which remains protected by its original copyright holder.^[17] Determining whether a work infringes on the original copyright hinges on the doctrine of substantial similarity, which requires that the derivative work copies protected expression from the original rather than unprotectable ideas. This principle was articulated in the landmark case Nichols v. Universal Pictures Corp. (1930), where Judge Learned Hand introduced the abstraction test to distinguish protectable expression from ideas. The abstraction-filtration-comparison test was later formalized in Computer Associates International, Inc. v. Altai, 982 F.2d 693 (2d Cir. 1992).^[18]^[19] To avoid infringement, derivative works must demonstrate sufficient transformation, such as altering narrative structure, characters, or themes in a way that creates meaningful new expression, often necessitating a license from the original copyright owner.^[20] For instance, the film adaptations of J.R.R. Tolkien's The Lord of the Rings novels, directed by Peter Jackson, were authorized through a 1969 licensing agreement between J.R.R. Tolkien and United Artists, with rights later managed by the Tolkien Estate, allowing the transformation of the literary works into motion pictures while preserving the estate's control over further derivations.^[21] Internationally, the Berne Convention for the Protection of Literary and Artistic Works (1886) harmonizes protections for derivative works by granting authors the exclusive right to authorize adaptations and translations of their works, ensuring reciprocity among member states. However, variations exist, particularly in European jurisdictions where moral rights—such as the right of integrity—allow authors to oppose derogatory or mutilating adaptations that harm their honor or reputation, even if commercially licensed. This right, rooted in Article 6bis of the Berne Convention and implemented in laws like France's Intellectual Property Code, provides stronger safeguards against unauthorized or unflattering derivations compared to the more economic-focused U.S. approach.

Derivation of Title in Property Law

In property law, derivation of title refers to the process of establishing legal ownership through a verifiable sequence of transfers, known as the chain of title, which traces the property's history from its original grant by a sovereign authority or acquisition via adverse possession to the current owner. This chain is documented through deeds, mortgages, wills, and other recorded instruments that demonstrate each conveyance's validity, ensuring that the title is marketable and free from undisclosed claims or defects.^[22]^[23] The concept serves as a foundational principle to prevent disputes by providing constructive notice of prior interests to subsequent purchasers.^[24] The historical roots of title derivation lie in English common law, where the need for clear proof of ownership emerged from feudal land tenure systems that required tracing rights back to the crown or original grants. In the United States, this evolved through colonial adoption and state recording statutes in the 19th century, which mandated public registries to preserve the chain and protect bona fide purchasers. A significant adaptation occurred with the introduction of the Torrens system in South Australia via the Real Property Act 1858, which streamlined derivation by issuing state-guaranteed certificates of title as conclusive evidence of ownership, thereby minimizing the necessity for exhaustive historical searches.^[25]^[26]^[27] Key principles governing derivation include the doctrine of after-acquired title, under which a grantor who conveys property without holding title at the time of transfer is estopped from later claiming it if they subsequently acquire ownership, with the title automatically passing to the original grantee. This estoppel by deed was affirmed in the U.S. Supreme Court case Van Rensselaer v. Kearney (1850), where the Court held that covenants in a deed bind the grantor to any later-obtained interest, ensuring continuity in the chain.^[28]^[29] In practice, derivation is validated through title searches conducted by attorneys or abstractors, which review public records to uncover transfers via sales, inheritances, foreclosures, or easements, identifying any breaks in the chain such as unrecorded liens. To mitigate risks from derivation defects, title insurance policies are commonly issued, indemnifying buyers against losses from hidden title flaws while the insurer assumes responsibility for curing issues. For instance, a search might reveal a foreclosure in 1920 that derived title to a subsequent heir, confirming the current owner's superior claim.^[23]^[30]^[31]

Music

Derived Rows in Twelve-Tone Technique

In Arnold Schoenberg's twelve-tone technique, developed in the early 1920s, a derived row refers to any transformation of the original prime row (P) generated through operations such as inversion, retrograde, or transposition, ensuring all twelve pitch classes are treated equally without establishing a tonal center.^[32] The prime row serves as the foundational series, an ordered arrangement of the twelve chromatic pitches, from which all other forms are systematically derived to maintain structural unity in atonal compositions.^[33] The derivation process begins with selecting a prime row, such as the one used in Schoenberg's Suite for Piano, Op. 25 (1921–1923): E–F–G–D♭–G♭–E♭–A♭–D–B–C–A–B♭ (or in pitch-class integers: 4, 5, 7, 1, 6, 3, 8, 2, 11, 0, 9, 10).^[34] To derive the inversion (I), the intervals between consecutive pitches are reversed in direction while preserving their size; for instance, the ascending minor second from E to F becomes a descending minor second from E to E♭, yielding I-4.^[32] The retrograde (R) simply reverses the order of the prime row, so R-4 would read B♭–A–C–B–D–A♭–E♭–G♭–D♭–G–F–E.^[33] The retrograde inversion (RI) combines these by inverting the retrograde or retrograding the inversion, producing RI-4.^[35] Each of these four basic forms (P, I, R, RI) can then be transposed to start (or end, for retrogrades) on any of the twelve chromatic pitches, resulting in up to 48 distinct row forms per prime row, though symmetries in some rows may reduce this number.^[33] Schoenberg employed derived rows extensively in his Suite for Piano, Op. 25 to achieve thematic cohesion, using forms like P-4, I-4, P-10, I-10, and their retrogrades partitioned into tetrachords for motivic development across movements such as the Prelude and Gigue.^[34] Later composers adopted this approach; for example, Igor Stravinsky incorporated derived rows in Threni (1958), his first fully twelve-tone composition, where transformations of a single row form the basis for choral and orchestral textures, marking his transition to serialism.^[36] Theoretically, derived rows promote combinatoriality, a property where subsets (such as hexachords) from different row forms combine to form aggregates containing all twelve pitches without repetition, reinforcing the equality of tones and suppressing tonal hierarchies.^[37] This ensures that musical elements like hexachords maintain the twelve-tone principle even when overlapped or segmented, providing a framework for coherent atonal structures.^[32]

Other Derivational Methods in Composition

In Baroque music, derivation often manifests through fugal techniques, where a primary subject generates accompanying countersubjects and subsequent contrapuntal lines. Johann Sebastian Bach's The Art of Fugue (BWV 1080, composed in the 1740s) exemplifies this by building an entire cycle of fourteen contrapuncti from a single d-minor subject, with countersubjects derived to complement the subject's harmonic and rhythmic profile while maintaining invertibility at the octave.^[38] Countersubjects emerge alongside the subject during expositions, evolving through stretto entries and episodic developments that fragment and recombine motivic elements for structural unity.^[39] This method roots in earlier contrapuntal practices but reaches a pinnacle in Bach's work, demonstrating exhaustive thematic transformation without altering the core subject's intervallic content.^[40] In the Classical and Romantic eras, thematic derivation shifted toward organic evolution within sonata forms, particularly through "developing variation," a technique where initial motifs undergo continuous intervallic, rhythmic, and textural modifications rather than strict repetition. Johannes Brahms mastered this in his symphonies, as seen in the Second Symphony (Op. 73, 1877), where the Adagio movement derives its entire structure from a descending motivic line in Theme Ia, extending variations across sections without exact restatements to create a sense of perpetual growth.^[41] Brahms' approach, later formalized by Arnold Schoenberg, emphasizes reinterpretation of brief motives—such as inverting a fourth to a fifth or displacing rhythms—to build thematic coherence, evident in works like the First Symphony (Op. 68, 1876), where primary themes spawn secondary ideas through subtle harmonic displacements.^[42] This contrasts with more static variation forms, prioritizing motivic progeny over ornamental embellishment for symphonic depth.^[41] Twentieth-century minimalism introduced derivation via repetitive processes that generate complexity from simplicity, notably through phase shifting, where offset repetitions of a motif yield emergent patterns. Steve Reich's Piano Phase (1967) for two pianos begins with both players repeating a twelve-note pattern in unison, but one gradually accelerates, creating phased offsets that derive interlocking rhythms—such as one player advancing by a sixteenth note after several repetitions—resulting in twelve distinct composite textures before shortening to eight- and four-note variants.^[43] This technique derives new musical entities from the initial motif's internal relationships, fostering auditory illusions of canon without traditional imitation, and influenced subsequent process-based compositions.^[43] In jazz improvisation, derivation frequently stems from harmonic frameworks like chord progressions, allowing solos to evolve organically from underlying cycles. John Coltrane's Giant Steps (1959) employs "Coltrane Changes," a progression modulating via major-third cycles (e.g., B major to G major to E-flat major) integrated with dominant seventh chords rooted in the circle of fifths, providing a scaffold for deriving scalar and intervallic lines in solos.^[44] Coltrane's own improvisation navigates these shifts by outlining ii-V-I patterns within each key center, transforming the "head" theme's motivic fragments into fluid, high-speed derivations that emphasize chromatic connectivity and modal interchange.^[44] This method extends Baroque and Classical principles into spontaneous creation, where the cycle of fifths serves as a generative source for endless variation in performance.^[45] These methods across genres highlight derivation's versatility in composition, evolving from rigid contrapuntal derivations in fugues to fluid, process-driven transformations, distinct from the systematic row manipulations in twelve-tone technique.^[43]

Science and Mathematics

Differentiation in Calculus

In calculus, differentiation is the process of finding the derivative of a function, which measures the instantaneous rate of change of the function with respect to its input variable. The derivative of a function f(x) at a point a, denoted f'(a), is formally defined as the limit

f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h},

provided this limit exists; geometrically, it represents the slope of the tangent line to the graph of f at x = a.^[46] This definition captures the notion of how the function's output changes as the input varies infinitesimally around a, distinguishing it from average rates of change over finite intervals.^[47] The foundational ideas of differentiation emerged in the late 17th century through the independent work of Isaac Newton and Gottfried Wilhelm Leibniz. Newton developed the method of fluxions around 1671, viewing derivatives as rates of change of "fluents" (quantities varying with time), which he applied to problems in physics and geometry.^[48] Leibniz introduced the differential notation \frac{dy}{dx} in 1675, emphasizing infinitesimals as small differences dy and dx whose ratio approximates the tangent slope, facilitating symbolic manipulation in calculus.^[49] Although their approaches lacked full rigor and sparked a priority dispute, they laid the groundwork for calculus; formal rigor was later provided by Augustin-Louis Cauchy in the 1820s, who defined limits precisely in his 1821 Cours d'analyse, enabling proofs based on epsilon-delta arguments.^[50] To illustrate the limit definition, consider the function f(x) = x^2. The derivative at a general point x is

f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h}.

Expanding the numerator gives (x + h)^2 - x^2 = x^2 + 2xh + h^2 - x^2 = 2xh + h^2, so

f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} \left(2x + h\right) = 2x,

after simplifying by dividing through by h (for h \neq 0) and taking the limit. This result shows that the slope of the tangent to y = x^2 at any x is $2x.^[46] One key application of differentiation arises in physics, where the derivative describes motion: if s(t) represents the position of an object at time t, then the velocity v(t) = s'(t) is the instantaneous rate of change of position, quantifying how quickly the object moves at a specific instant rather than over an interval. For instance, under constant acceleration like gravity, this ties into classical mechanics by linking position changes to forces, though full equations require integration for position from velocity.^[51]

Logical Derivation

In formal logic, a derivation, often synonymous with a proof, constitutes a finite sequence of well-formed formulas wherein each formula is either an axiom of the system, an explicit assumption (premise), or logically follows from preceding formulas via a designated inference rule, such as modus ponens, which allows the inference of Q from P \to Q and P.^[52] This structure ensures that derivations systematically build conclusions from foundational elements, providing a mechanical verification of logical validity within a given formal system.^[53] Prominent proof systems for conducting derivations include Hilbert-style systems and natural deduction. Hilbert-style systems, developed by David Hilbert and collaborators, emphasize a minimal set of axioms capturing logical truths combined with modus ponens as the primary inference rule, enabling derivations through repeated applications to generate theorems.^[54] In contrast, natural deduction systems, introduced by Gerhard Gentzen in 1935, employ pairs of introduction and elimination rules tailored to each logical connective—such as conjunction introduction (from P and Q, derive P \land Q) and elimination (from P \land Q, derive P)—facilitating derivations that mirror intuitive reasoning patterns.^[55] A simple example in propositional logic using modus ponens within a Hilbert-style framework proceeds as follows: assume premises P \to Q (line 1) and P (line 2); then, by modus ponens, derive Q (line 3). This illustrates how derivations enforce strict adherence to rules, yielding only consequences entailed by the premises.^[56] Historically, derivations trace back to Euclid's Elements around 300 BCE, which organized geometric knowledge through axiomatic deductions, deriving theorems from primitive notions and postulates via implicit inference steps, establishing a paradigm for systematic proof in mathematics.^[57] A pivotal modern advancement came with Kurt Gödel's 1930 completeness theorem, which demonstrated that in first-order predicate logic, every semantically valid formula—true in all models—is syntactically derivable from the axioms using the system's rules.^[58] Logical systems exhibiting soundness and completeness possess complementary properties essential for their reliability: soundness guarantees that every derivable formula is semantically valid (true under all interpretations satisfying the premises), preventing the proof of falsehoods; completeness ensures the converse, that every valid formula is derivable, capturing all logical truths.^[59] These properties hold jointly for standard first-order logic, as proven by Gödel, affirming the equivalence between syntactic derivability and semantic entailment in this framework.^[60]

Derivations in Differential Algebra

In differential algebra, a derivation on a unital ring R is an additive map D: R \to R satisfying the Leibniz rule D(ab) = a D(b) + D(a) b for all a, b \in R, with the additional property that D(1) = 0. The definition includes the condition D(1) = 0. This follows from the Leibniz rule in rings of characteristic not equal to 2, since D(1) = D(1 \cdot 1) = 1 \cdot D(1) + D(1) \cdot 1 = 2 D(1), implying D(1) = 0. In characteristic 2, the condition is imposed explicitly as part of the definition to ensure consistency, particularly under k-linearity.^[61]^[62] This structure generalizes the familiar derivative from calculus to abstract rings, without relying on analytic limits, and extends naturally to non-commutative settings.^[61] A concrete example arises in the polynomial ring k[x, y] over a field k, where the partial derivative \partial / \partial x acts as a derivation by \partial / \partial x \left( \sum a_{ij} x^i y^j \right) = \sum i a_{ij} x^{i-1} y^j.^[61] For instance, \partial / \partial x (x^2 y) = 2 x y.^[61] Such derivations capture infinitesimal changes in algebraic varieties, motivating their role in broader structures. Derivations find key applications in algebraic geometry, where the tangent space to a variety at a point corresponds to the vector space of k-derivations from the localized coordinate ring to the residue field at that point.^[63] This perspective aligns derivations with geometric intuition, treating them as tangent vectors that probe first-order behavior along curves through the point.^[63] The foundations of differential algebra, including systematic study of derivation-based polynomial ideals and elimination methods, were established by Joseph Ritt in his seminal 1950 monograph.^[64] The collection \operatorname{Der}(R) of all derivations on R forms a Lie algebra under the commutator bracket [D_1, D_2] = D_1 \circ D_2 - D_2 \circ D_1.^[65] For simple rings, such as the matrix algebra R = M_n(k) over an algebraically closed field k of characteristic zero, all derivations are inner (of the form D_A(X) = A X - X A for some A \in R), and \operatorname{Der}(R) is isomorphic to the special linear Lie algebra \mathfrak{sl}_n(k), which has dimension n^2 - [1](/page/1).^[65] This dimension reflects the trace-zero condition arising from the center of scalars.^[65]

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Row Properties – Open Music Theory - VIVA's Pressbooks
While discussion of derived rows typically focuses on subsegments of three or four notes, the two discrete hexachords of a row (i.e., the two halves, pitches 1– ...
[39]
The Art of The Fugue | Orchestra of St. Luke's
The Art of The Fugue was written at the end of Bach's life, and was a compendium intended to demonstrate all the possibilities of contrapuntal writing.
[40]
High Baroque Fugal Exposition – Open Music Theory
This chapter deals with the basics of writing a fugal exposition from a given subject according to the “high Baroque” technique of Bach, Handel, and their ...
[41]
[PDF] The Art of Fugue - OAPEN Library
“Bach's Use of Hitherto Unrecognized Types of. Countersubjects in the 'Art of Fugue.' ” Journal of the American Musi- cological Society 31 (1978): 344–62 ...
[42]
[PDF] Developing Variations - Uppsala universitet
The relationship between these clusters can be explained by developing variation which Brahms uses to alter the themes. This organic relationship within the ...
[43]
Brahms's First Symphony: Thematic Coherence and Its Secret Origin
An outline of its hist provides the necessary background to such speculation. Brahms made only two references to the Symphony including mu examples. The first ...
[44]
[PDF] A Performer's Guide to the Phase Music of Steve Reich - eScholarship
He would later come to call this his “phase-shifting process.” Reich used this technique with tape further in his pieces Come Out and Melodica.Missing: derivation | Show results with:derivation<|separator|>
[45]
Coltrane Changes Explained - TJPS - The Jazz Piano Site
Coltrane Changes are a chord progression that changes keys by Major 3rd intervals. Coltrane didn't invent this idea but was the first to use it extensively.
[46]
Jazz Deconstructed: What Makes John Coltrane's "Giant Steps" So ...
Nov 13, 2018 · The title track of Giant Steps has become known for a rapid chord progression that cycles through three keys, built on an earlier technique ...
[47]
Calculus I - The Definition of the Derivative - Pauls Online Math Notes
Nov 16, 2022 · This is such an important limit and it arises in so many places that we give it a name. We call it a derivative. Here is the official definition ...
[48]
Derivatives Using the Limit Definition - UC Davis Mathematics
The following problems require the use of the limit definition of a derivative, which is given by tex2html_wrap_inline338. They range in difficulty from easy ...
[49]
Isaac Newton - Stanford Encyclopedia of Philosophy
Dec 19, 2007 · Isaac Newton (1642–1727) is best known for having invented the calculus in the mid to late 1660s (most of a decade before Leibniz did so ...
[50]
[PDF] 14. Calculus of Several Variables
The notation df/dx was introduced by Leibniz in 1675, and reflects his infinitesimal approach to derivatives. 1 The Swiss mathematician Leonhard Euler (1707– ...Missing: differentiation | Show results with:differentiation
[51]
The origin of Cauchy's conceptions of a definite integral and of the ...
The hypothesis of a function being o continuous ) in the modern sense is made in theorems in the later part of this memoir (I). XII. In October, 1821, CAUCHY ...
[52]
Derivatives | Engineering Math Resource Center
Velocity and acceleration are both rates of change of position over time. Namely, velocity (v) is the derivative of position (s), and acceleration (a) is ...
[53]
[PDF] Sentential Logic Primer - Rice University
Jul 23, 2004 · (42) DEFINITION: A derivation of the formula ϕ is a derivation with the fol- lowing properties. (a) The derivation ends with the line “ϕ ”.<|separator|>
[54]
[PDF] An Introduction to Proof Theory - UCSD Math
This introductory chapter will deal primarily with the sequent calculus, and resolution, and to lesser extent, the Hilbert-style proof systems and the natural.
[55]
[PDF] Foundations of Mathematics Vol. 1 (1934) David Hilbert/Paul Bernays
To attack the problem in these terms we must first aim at an overview of the possible logical inferences that can be made from an axiom system. The ...
[56]
[PDF] I shall develop the calculus of natural deduction in section II of this ...
In section III of this paper, therefore, I shall develop a new calculus of logical deduction possessing all the desired properties in both their intui- tionist ...
[57]
[PDF] Formal Proof Systems for First-Order Logic | Computer Science
or ...
[58]
[PDF] Euclid and beyond: towards a long-term history of deductivity
The deductive organization of Euclid's Elements serves as a model for mathematical and scientific texts in a variety of subjects from antiquity through the ...
[59]
[PDF] Die Vollst~ndigkeit der Axiome des logischen Funktionenkalkiils ~).
Von Kurt GSdel in Wien. Whitehead und Russell haben bekanntlich die Logik und. Mathematik so aufgebaut, datl sie gewisse evidente S~ttze als Axiome.Missing: Gödel | Show results with:Gödel
[60]
Logical Consequence and First-Order Soundness and Completeness
Abstract. What is the philosophical significance of the soundness and completeness theorems for first-order logic? In the first section of this paper I raise ...
[61]
[PDF] Soundness and Completeness Proofs by Coinductive Methods
Soundness is often easy to prove, but it becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several ...
[62]
[PDF] Kähler Differentials from a Geometric Point of View - Brown Math
Apr 10, 2019 · To see this, note that D(1) = D(1 · 1) = 1D(1)+1D(1) by the Leibniz rule, so that D(1) = 0. It then follows from k-linearity that D(a) = aD(1) = ...
[63]
[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 21
Jan 11, 2008 · In differential geometry, the tangent space at a point is sometimes defined as the vector space of derivations at that point. A derivation ...
[64]
Differential Algebra - AMS Bookstore - American Mathematical Society
Ritt and his collaborators in the 1930's was to give the classical theory of nonlinear differential equations, similar to the theory created by Emmy Noether and ...
[65]
[PDF] Semi-Simple Algebras and Generalized Derivations
Roughly speaking, a derivation of an algebra is the infini- tesimal operation corresponding to an automorphism. This notion plays an important part in the study ...