Английская Википедия:Glossary of calculus

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Шаблон:Short description Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones.

This glossary of calculus is a list of definitions about calculus, its sub-disciplines, and related fields.

F

Шаблон:Term Шаблон:Defn\cdot f^{(m_1+\cdots+m_n)}(g(x))\cdot \prod_{j=1}^n\left(g^{(j)}(x)\right)^{m_j},</math>

where the sum is over all n-tuples of nonnegative integers (m₁, …, m_n) satisfying the constraint

<math>1\cdot m_1+2\cdot m_2+3\cdot m_3+\cdots+n\cdot m_n=n.</math>

Sometimes, to give it a memorable pattern, it is written in a way in which the coefficients that have the combinatorial interpretation discussed below are less explicit:

<math>{d^n \over dx^n} f(g(x))

=\sum \frac{n!}{m_1!\,m_2!\,\cdots\,m_n!}\cdot f^{(m_1+\cdots+m_n)}(g(x))\cdot \prod_{j=1}^n\left(\frac{g^{(j)}(x)}{j!}\right)^{m_j}.</math>

Combining the terms with the same value of m₁ + m₂ + ... + m_n = k and noticing that m _j has to be zero for j > n − k + 1 leads to a somewhat simpler formula expressed in terms of Bell polynomials B_n,k(x₁,...,x_n−k+1):

<math>{d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g(x),\dots,g^{(n-k+1)}(x)\right).</math>}}

I

Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn{d\mathit{t}} .</math>

From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (Шаблон:Math vs. Шаблон:Math graph) is the displacement, Шаблон:Math. In calculus terms, the integral of the velocity function Шаблон:Math is the displacement function Шаблон:Math. In the figure, this corresponds to the yellow area under the curve labeled Шаблон:Math (Шаблон:Math being an alternative notation for displacement).

<math>\boldsymbol{x} = \int \boldsymbol{v} \ d\mathit{t} .</math>

Since the derivative of the position with respect to time gives the change in position (in metres) divided by the change in time (in seconds), velocity is measured in metres per second (m/s). Although the concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment. .}} Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn Шаблон:Term Шаблон:Defn

See also

• Outline of calculus
• Glossary of areas of mathematics
• Glossary of astronomy
• Glossary of biology
• Glossary of botany
• Glossary of chemistry
• Glossary of ecology
• Glossary of engineering
• Glossary of physics
• Glossary of probability and statistics

References

Works cited

• Шаблон:Citation.
• Шаблон:Citation.
• Шаблон:Cite book
• Шаблон:Citation.
• Шаблон:Cite book
• Шаблон:Citation.