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total_derivative [2015/03/28 16:28] nikolaj |
total_derivative [2015/03/28 16:36] nikolaj |
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| ===== Total derivative ===== | ===== Total derivative ===== | ||
| ==== Function ==== | ==== Function ==== | ||
| - | | @#FF9944: definition | @#FF9944: $\dfrac{{\mathrm d}}{{\mathrm d}t}:\left(X_1\times\cdots \times X_n\times{\mathbb R}\to{\mathbb R}\right)\to \left({\mathbb R}\to X_1\times\cdots \times X_n\right)\times \left({\mathbb R}\to {\mathbb R}\right)$ | | + | | @#FF9944: definition | @#FF9944: $\dfrac{{\mathrm d}}{{\mathrm d}t}:\left(X_1\times\cdots \times X_n\times{\mathbb R}\to{\mathbb R}\right)\to \left(\left({\mathbb R}\to X_1\times\cdots \times X_n\right)\times {\mathbb R}\right)\to {\mathbb R}$ | |
| | @#BBDDEE: let | @#BBDDEE: $\diamond\ f(x^1,\dots,x^n,t)$ | | | @#BBDDEE: let | @#BBDDEE: $\diamond\ f(x^1,\dots,x^n,t)$ | | ||
| | @#FF9944: definition | @#FF9944: $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)\left(t\mapsto\langle r^1(t),\dots,r^n(t)\rangle,t\right):=\sum_{j=1}^n \dfrac{\partial f}{\partial x^j}(\langle r^1(t),\dots,r^n(t),t\rangle)\cdot\dfrac{\partial r^j}{\partial t}(t)+\dfrac{\partial f}{\partial t}(\langle r^1(t),\dots,r^n(t),t\rangle)$ | | | @#FF9944: definition | @#FF9944: $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)\left(t\mapsto\langle r^1(t),\dots,r^n(t)\rangle,t\right):=\sum_{j=1}^n \dfrac{\partial f}{\partial x^j}(\langle r^1(t),\dots,r^n(t),t\rangle)\cdot\dfrac{\partial r^j}{\partial t}(t)+\dfrac{\partial f}{\partial t}(\langle r^1(t),\dots,r^n(t),t\rangle)$ | | ||
