basic_tools:calculus:leibniz_integration_formula
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basic_tools:calculus:leibniz_integration_formula [2018/03/10 17:53] iiqof created |
basic_tools:calculus:leibniz_integration_formula [2018/03/28 12:28] (current) jakobadmin |
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| ====== Leibniz Integration Formula ====== | ====== Leibniz Integration Formula ====== | ||
| + | <tabbox Intuitive> | ||
| + | <tabbox Concrete> | ||
| This tool gives a nice formula to the derivative of a one dimensional integral with dependencies to the varied quantity every where, with $f, a$ and $b$ having the right conditions | This tool gives a nice formula to the derivative of a one dimensional integral with dependencies to the varied quantity every where, with $f, a$ and $b$ having the right conditions | ||
| $$ | $$ | ||
| - | \left.\frac{\partial}{\partial \varepsilon} \int_{a(\varepsilon)}^{b(\varepsilon)}f(\varepsilon, x) d x \right|_{\varepsilon=0} = \int_{a}^{b} \left.\frac{\partial}{\partial \varepsilon} f(\varepsilon, x)\right|_{\varepsilon=0} d x + f(0,b)\left.\frac{\partial}{\partial \varepsilon} b(\varepsilon)\right|_{\varepsilon=0} - f(0,a)\left.\frac{\partial}{\partial \varepsilon} a(\varepsilon)\right|_{\varepsilon=0} | + | \left.\frac{\partial}{\partial \varepsilon} \int_{a(\varepsilon)}^{b(\varepsilon)}f(\varepsilon, x) d x \right|_{\varepsilon=0} = \\ \int_{a}^{b} \left.\frac{\partial}{\partial \varepsilon} f(\varepsilon, x)\right|_{\varepsilon=0} d x + f(0,b)\left.\frac{\partial}{\partial \varepsilon} b(\varepsilon)\right|_{\varepsilon=0} - f(0,a)\left.\frac{\partial}{\partial \varepsilon} a(\varepsilon)\right|_{\varepsilon=0} |
| $$ | $$ | ||
| + | <tabbox Abstract> | ||
| + | <tabbox Why is it interesting?> | ||
| - | ** Uses and further concepts** | + | This formula is really nice in one dimenisonal [[basic tools:variational calculus|Variational Calculus]] and |
| + | with $a(0) = a$, $b(0) = b$, and for the particle mechanics [[theorems:noethers_theorems|Noether's Theorem]]. For integrals over arbitrary manifolds, it generalizes to the [[start|Lie derivative]] of a volume element. | ||
| - | This formula is really nice on one dimenisonal [[basic tools:variational calculus|Variational Calculus]] and | + | </tabbox> |
| - | with $a(0) = a$, $b(0) = b$, and for the particle mechanics [[advanced_tools:noethers_theorems|Noether's Theorem]]. For integrals over arbitrary manifolds, it generalizes to the [[|Lie derivative]] of a volume element. | + | |
basic_tools/calculus/leibniz_integration_formula.1520700808.txt.gz · Last modified: 2018/03/10 16:53 (external edit)
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