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total_derivative [2015/03/28 16:36]
nikolaj
total_derivative [2016/03/11 15:14] (current)
nikolaj
Line 3: Line 3:
 | @#FF9944: definition ​ | @#FF9944: $\dfrac{{\mathrm d}}{{\mathrm d}t}:​\left(X_1\times\cdots \times X_n\times{\mathbb R}\to{\mathbb R}\right)\to \left(\left({\mathbb R}\to X_1\times\cdots \times X_n\right)\times {\mathbb R}\right)\to {\mathbb R}$ | | @#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: $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)\left(\langle ​t\mapsto\langle r^1(t),​\dots,​r^n(t)\rangle,​t\rangle\right):​=\sum_{j=1}^n \dfrac{\partial f}{\partial x^j}(\langle r^1(t),​\dots,​r^n(t),​t\rangle)\cdot\dfrac{\partial r^j}{\partial t}(t)+\dfrac{\partial f}{\partial t}(\langle r^1(t),​\dots,​r^n(t),​t\rangle)$ |
  
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