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total_derivative [2015/03/28 16:28] nikolaj |
total_derivative [2016/03/11 15:14] 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: $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)\left(\langle t\mapsto\langle r^1(t),\dots,r^n(t)\rangle,t\rangle\right):=\sum_{j=1}^n \dfrac{\partial f}{\partial x^j}(\langle r^1(t),\dots,r^n(t),t\rangle)\cdot\dfrac{\partial r^j}{\partial t}(t)+\dfrac{\partial f}{\partial t}(\langle r^1(t),\dots,r^n(t),t\rangle)$ | |
| ----- | ----- | ||
| === Example === | === Example === | ||
| - | The total derivative of the function | + | The total derivative of the (not explicitly time dependent) function |
| $f:{\mathbb R}^3\to{\mathbb R}, \ \ \ f(x,y,t):=x^2\cos(y)$ | $f:{\mathbb R}^3\to{\mathbb R}, \ \ \ f(x,y,t):=x^2\cos(y)$ | ||
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| $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)(R,t)=2\,R_x(t)\cos(R_y(t))\cdot R_x'(t)+R_x(t)^2\sin(R_y(t))\cdot R_y'(t)$ | $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)(R,t)=2\,R_x(t)\cos(R_y(t))\cdot R_x'(t)+R_x(t)^2\sin(R_y(t))\cdot R_y'(t)$ | ||
| - | $=2\,7^2\,t\cos(-3t^5)-3\,5\,7^2)\,t^{2+4}\sin(-3t^5)$ | + | $=2\cdot 7^2\,t\cos(-3t^5)-3\cdot 5\cdot 7^2\,t^{2+4}\sin(-3t^5)$ |
| === Reference === | === Reference === | ||
