- орт внешней нормали, направление наиболее быстрого возрастания скаляра v. - набла, или оператор Гамильтона, который можно рассматривать как вектор. Под оператором понимается совокупность математических действий, в данном случае дифференцирование. Сам по себе оператор ничего не означает, он имеет смысл, когда применим к какой-либо величине.
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в декартовой системе координат.
Градиент является дифференциальной характеристикой скалярного поля.
Пространственное изменение скаляра вдоль направления L:
- пространственная скорость возрастания vв направлении равна проекции градиента v на это направление.
Заметим, скалярное поле v порождает векторное поле =![](data:image/png;base64,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) 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