winding number (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun

Phonetic Transcription

/wˈaɪndɪŋ nʌmbər/

Meaning

The term "winding number" refers to a concept in mathematics, specifically in topology and complex analysis. It quantifies the number of times a curve winds around a particular point in the plane. It is an important concept for understanding topological properties and is widely used in various branches of mathematics, including algebraic topology and complex functions.

• Usage: The term is primarily used in written contexts, particularly in academic and educational materials, such as textbooks and research papers in mathematics and physics.
• Frequency of Use: It is a specialized term and, while not commonly encountered in everyday language, is frequently used in professional and academic circles.

Example Sentences

1. The winding number of a closed curve around a point determines whether it encloses the point in the complex plane.

2. Вinding число замкнутой кривой вокруг точки определяет, замыкает ли она точку в комплексной плоскости.

3. In topology, the concept of a winding number helps classify different types of curves based on their behavior around singular points.

4. В топологии концепция winding number помогает классифицировать различные типы кривых на основе их поведения вокруг сингулярных точек.

5. One can compute the winding number by integrating the derivative of a complex-valued function around the given curve.

6. Winding number можно вычислить, интегрируя производную комплекснозначной функции вокруг данной кривой.

Idiomatic Expressions

While "winding number" itself does not have idiomatic expressions, it is often involved in mathematical discussions where it complements other concepts:

1. "To wrap around" can be used in discussions related to winding number.
2. The path you take wraps around the origin multiple times.

3. Путь, который вы выбрали, обвивает начало много раз.

4. "Circle back" is another phrase that may appear in discussions about curves and winding numbers.

5. In the analysis, we need to circle back to the point where the winding number is first defined.

6. В анализе нам нужно вернуться к точке, где первое определено winding число.

7. "Go around in circles" when illustrating concepts involving winding numbers and their application in complex analysis.

8. Sometimes our proofs make us feel like we are going around in circles without reaching the conclusion about the winding number.
9. Иногда наши доказательства заставляют нас чувствовать, что мы кружим, не достигая вывода о winding числе.

Etymology

The term "winding number" comes from the combination of "winding," deriving from the Old English "windan," which means "to twist or turn," and "number," from Latin "numerus," which indicates a mathematical quantity. The phrase has been used in various mathematical contexts to imply the way a figure wraps around another point.

Synonyms and Antonyms

• Synonyms:
• Topological index

• Loop count

• Antonyms:

• No direct antonyms, but concepts like "unwound" or "linear" may imply curves that do not wrap around points.

This detailed analysis of "winding number" should provide a comprehensive understanding of its usage and relevance in mathematical contexts.