Part of Speech

Noun phrase

Phonetic Transcription

/ˈtɜrmˌwaɪz dɪˌfɜrʃiəˈbɪlɪti/

Meaning and Usage

Termwise differentiability refers to the property of a series of functions where the differentiation of each function in the series can be performed term by term, under certain convergence conditions. This concept is primarily used in calculus and functional analysis, especially in the context of power series and series of functions.

The usage of "termwise differentiability" is relatively specialized and is more often found in written academic contexts than in everyday speech. It may appear in mathematical texts, research papers, and advanced mathematical discussions.

Example Sentences

1. The theorem states that under specific conditions, the series is termwise differentiable, allowing us to differentiate the series as a whole.

2. Теорема утверждает, что при определенных условиях последовательность можно дифференцировать по терминам, что позволяет нам дифференцировать всю последовательность.

3. To establish termwise differentiability of the given power series, we need to examine its radius of convergence carefully.

4. Чтобы установить дифференцируемость по терминам данной степенной серии, нам необходимо внимательно изучить ее радиус сходимости.

5. The importance of termwise differentiability lies in its application to solving differential equations involving series.

6. Важность дифференцируемости по терминам заключается в ее применении к решению дифференциальных уравнений, связанных с рядами.

Idiomatic Expressions

The term "termwise" itself may not be commonly found in idiomatic expressions. However, in the context of mathematical discussions, the concept of differentiability often coexists with phrases that indicate the process or nature of change in functions or series. Here are some examples:

1. "Differentiation under the integral sign" is a useful tool that complements termwise differentiability.

2. "Дифференцирование под знаком интеграла" является полезным инструментом, который дополняет дифференцируемость по терминам.

3. "Pointwise convergence" is often compared with termwise differentiability in mathematical analysis.

4. "Почти сходимость" часто сравнивается с дифференцируемостью по терминам в математическом анализе.

5. "Continuity of mapping" is crucial when examining conditions for termwise differentiability.

6. "Непрерывность отображения" имеет решающее значение при изучении условий для дифференцируемости по терминам.

Etymology

The term "termwise differentiability" is composed of "termwise," derived from the combination of "term," referring to each individual part or element, and "wise," indicating manner or method; and "differentiability," which derives from the word "differentiate," coming from the Latin root "differentiāre," meaning to set apart or to distinguish.

Synonyms and Antonyms

Synonyms: - Differentiable series - Term-by-term differentiable

Antonyms: - Non-differentiable series - Indifferentiable series

This framework provides a comprehensive overview of "termwise differentiability," its meanings, usages, and related concepts in the mathematical context.