See differential operator in All languages combined, or Wiktionary
{
"forms": [
{
"form": "differential operators",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "differential operator (plural differential operators)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"categories": [
{
"kind": "other",
"name": "English entries with incorrect language header",
"parents": [
"Entries with incorrect language header",
"Entry maintenance"
],
"source": "w"
},
{
"kind": "other",
"name": "Entries with translation boxes",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with 1 entry",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Pages with entries",
"parents": [],
"source": "w"
},
{
"kind": "other",
"name": "Terms with Mandarin translations",
"parents": [],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Functions",
"orig": "en:Functions",
"parents": [
"Algebra",
"Calculus",
"Geometry",
"Mathematical analysis",
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematical analysis",
"orig": "en:Mathematical analysis",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
},
{
"kind": "topical",
"langcode": "en",
"name": "Mathematics",
"orig": "en:Mathematics",
"parents": [
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w"
}
],
"examples": [
{
"text": "Define P(D) to be the differential operator given by P(D)#61;D²#43;5D#43;6, where D is the differentiation operator. Then P(D)(#92;sin(x))#61;-#92;sin(x)#43;5#92;cos(x)#43;6.",
"type": "example"
},
{
"bold_text_offsets": [
[
66,
88
]
],
"ref": "2000, S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, page 328:",
"text": "[…]these conditions appeared in the very first papers on ordinary differential operators.",
"type": "quote"
},
{
"bold_text_offsets": [
[
2,
23
],
[
351,
372
],
[
351,
373
]
],
"ref": "2012, Youri Egorov, Bert-Wolfgang Schulze, Pseudo-Differential Operators, Singularities, Applications, page 27:",
"text": "A differential operator P#92;left(D#92;right) is hypoelliptic if for any domain ⊂ #92;mathbb#123;R#125;ⁿ any solution u of the equation P#92;left(D#92;right)u#61;0 from the class #92;mathfrak#123;D#125;'#92;left(#92;Omega#92;right) is a function from C#92;infty(ω) for any open set ω ⊂⊂ #92;Omega.\nA complete algebraic description of all hypoelliptic differential operators has been obtained by Hörmander in [H1].",
"type": "quote"
},
{
"bold_text_offsets": [
[
10,
32
]
],
"ref": "1998, L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, page 251:",
"text": "Secondly, differential operators and to some extent their fundamental solutions are local even with respect to the wave front set.",
"type": "quote"
}
],
"glosses": [
"An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives)."
],
"hyponyms": [
{
"word": "Laplace operator"
},
{
"word": "Laplacian"
},
{
"word": "d'Alembert operator"
},
{
"word": "d'Alembertian"
}
],
"id": "en-differential_operator-en-noun-WBmqaSXo",
"links": [
[
"mathematics",
"mathematics"
],
[
"mathematical analysis",
"mathematical analysis"
],
[
"operator",
"operator"
],
[
"function",
"function"
],
[
"differentiation",
"differentiation"
],
[
"derivative",
"derivative#English"
]
],
"raw_glosses": [
"(mathematics, mathematical analysis) An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives)."
],
"related": [
{
"word": "partial differential operator"
},
{
"word": "pseudodifferential operator"
}
],
"topics": [
"mathematical-analysis",
"mathematics",
"sciences"
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "wēifēn suànzǐ",
"sense": "mathematics: an operator defined as a function of the differentiation operator",
"word": "微分算子"
}
],
"wikipedia": [
"differential operator"
]
}
],
"word": "differential operator"
}
{
"forms": [
{
"form": "differential operators",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "differential operator (plural differential operators)",
"name": "en-noun"
}
],
"hyponyms": [
{
"word": "Laplace operator"
},
{
"word": "Laplacian"
},
{
"word": "d'Alembert operator"
},
{
"word": "d'Alembertian"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "partial differential operator"
},
{
"word": "pseudodifferential operator"
}
],
"senses": [
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English lemmas",
"English multiword terms",
"English nouns",
"English terms with quotations",
"English terms with usage examples",
"Entries with translation boxes",
"Pages with 1 entry",
"Pages with entries",
"Quotation templates to be cleaned",
"Terms with Mandarin translations",
"en:Functions",
"en:Mathematical analysis",
"en:Mathematics"
],
"examples": [
{
"text": "Define P(D) to be the differential operator given by P(D)#61;D²#43;5D#43;6, where D is the differentiation operator. Then P(D)(#92;sin(x))#61;-#92;sin(x)#43;5#92;cos(x)#43;6.",
"type": "example"
},
{
"bold_text_offsets": [
[
66,
88
]
],
"ref": "2000, S. Albeverio, P. Kurasov, Singular Perturbations of Differential Operators, page 328:",
"text": "[…]these conditions appeared in the very first papers on ordinary differential operators.",
"type": "quote"
},
{
"bold_text_offsets": [
[
2,
23
],
[
351,
372
],
[
351,
373
]
],
"ref": "2012, Youri Egorov, Bert-Wolfgang Schulze, Pseudo-Differential Operators, Singularities, Applications, page 27:",
"text": "A differential operator P#92;left(D#92;right) is hypoelliptic if for any domain ⊂ #92;mathbb#123;R#125;ⁿ any solution u of the equation P#92;left(D#92;right)u#61;0 from the class #92;mathfrak#123;D#125;'#92;left(#92;Omega#92;right) is a function from C#92;infty(ω) for any open set ω ⊂⊂ #92;Omega.\nA complete algebraic description of all hypoelliptic differential operators has been obtained by Hörmander in [H1].",
"type": "quote"
},
{
"bold_text_offsets": [
[
10,
32
]
],
"ref": "1998, L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, page 251:",
"text": "Secondly, differential operators and to some extent their fundamental solutions are local even with respect to the wave front set.",
"type": "quote"
}
],
"glosses": [
"An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives)."
],
"links": [
[
"mathematics",
"mathematics"
],
[
"mathematical analysis",
"mathematical analysis"
],
[
"operator",
"operator"
],
[
"function",
"function"
],
[
"differentiation",
"differentiation"
],
[
"derivative",
"derivative#English"
]
],
"raw_glosses": [
"(mathematics, mathematical analysis) An operator defined as a function of the differentiation operator (the operator which maps functions to their derivatives)."
],
"topics": [
"mathematical-analysis",
"mathematics",
"sciences"
],
"wikipedia": [
"differential operator"
]
}
],
"translations": [
{
"code": "cmn",
"lang": "Chinese Mandarin",
"roman": "wēifēn suànzǐ",
"sense": "mathematics: an operator defined as a function of the differentiation operator",
"word": "微分算子"
}
],
"word": "differential operator"
}
Download raw JSONL data for differential operator meaning in English (3.2kB)
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2025-04-29 from the enwiktionary dump dated 2025-04-20 using wiktextract (4eaa824 and ea19a0a). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.