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3次元スカラー/ベクトル関数の導関数 📂数理物理学

3次元スカラー/ベクトル関数の導関数

定理

3次元のスカラー関数f:R3R1f : \mathbb{R}^{3} \to \mathbb{R}^{1}f(x(t),y(t),z(t))=ff(x(t), y(t), z(t)) = fのとき、dfdt\dfrac{df}{dt}は次のようになる。

dfdt=fxdxdt+fydydt+fzdzdt \dfrac{d f}{d t} = \dfrac{\partial f}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial f}{\partial y}\dfrac{dy}{dt} + \dfrac{\partial f}{\partial z}\dfrac{dz}{dt}

3次元のベクター関数f:R3R3\mathbf{f} : \mathbb{R}^{3} \to \mathbb{R}^{3}f(x(t),y(t),z(t))=(f1,f2,f3)\mathbf{f}(x(t), y(t), z(t)) = (f_{1}, f_{2}, f_{3})のとき、dfdt\dfrac{d \mathbf{f}}{dt}は次のようになる。

dfdt= (df1dt,df2dt,df3dt)=df1dtx^+df2dty^+df3dtz^= (f1xdxdt+f1ydydt+f1zdzdt)x^+(f2xdxdt+f2ydydt+f2zdzdt)y^+(f3xdxdt+f3ydydt+f3zdzdt)z^ \begin{align*} \dfrac{d \mathbf{f}}{d t} =&\ \left( \dfrac{d f_{1}}{d t}, \dfrac{d f_{2}}{d t}, \dfrac{d f_{3}}{d t} \right) = \dfrac{d f_{1}}{d t}\hat{\mathbf{x}} + \dfrac{d f_{2}}{d t}\hat{\mathbf{y}} + \dfrac{d f_{3}}{d t}\hat{\mathbf{z}} \\[1em] =&\ \left( \dfrac{\partial f_{1}}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial f_{1}}{\partial y}\dfrac{dy}{dt} + \dfrac{\partial f_{1}}{\partial z}\dfrac{dz}{dt} \right)\hat{\mathbf{x}} + \left( \dfrac{\partial f_{2}}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial f_{2}}{\partial y}\dfrac{dy}{dt} + \dfrac{\partial f_{2}}{\partial z}\dfrac{dz}{dt} \right)\hat{\mathbf{y}} \\[1em] &+ \left( \dfrac{\partial f_{3}}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial f_{3}}{\partial y}\dfrac{dy}{dt} + \dfrac{\partial f_{3}}{\partial z}\dfrac{dz}{dt} \right)\hat{\mathbf{z}} \end{align*}

説明

多変数のベクター関数f:RnRm\mathbf{f} : \mathbb{R}^{n} \to \mathbb{R}^{m}f=(f1,f2,,fm)\mathbf{f} = \left( f_{1}, f_{2}, \dots, f_{m} \right)のとき、全微分は次のようになる。

f=[D1f1D2f1Dnf1D1f2D2f2Dnf2D1fmD2fmDnfm] \mathbf{f}^{\prime} = \begin{bmatrix} D_{1}f_{1} & D_{2}f_{1} & \cdots & D_{n}f_{1} \\ D_{1}f_{2} & D_{2}f_{2} & \cdots & D_{n}f_{2} \\ \vdots & \vdots & \ddots & \vdots \\ D_{1}f_{m} & D_{2}f_{m} & \cdots & D_{n}f_{m} \end{bmatrix}

だからn=3n=3m=1,3m=1,3のときは、次のようになる。

f=[D1fD2fD3f]=[fxfyfz] f^{\prime} = \begin{bmatrix} D_{1}f & D_{2}f & D_{3}f \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f}{\partial x} & \dfrac{\partial f}{\partial y} & \dfrac{\partial f}{\partial z} \end{bmatrix}

f=[D1f1D2f1D3f1D1f2D2f2D3f2D1f3D2f3D3f3]=[f1xf1yf1zf2xf2yf2zf3xf3yf3z] \mathbf{f}^{\prime} = \begin{bmatrix} D_{1}f_{1} & D_{2}f_{1} & D_{3}f_{1} \\ D_{1}f_{2} & D_{2}f_{2} & D_{3}f_{2} \\ D_{1}f_{3} & D_{2}f_{3} & D_{3}f_{3} \end{bmatrix} = \begin{bmatrix}\dfrac{\partial f_{1}}{\partial x} & \dfrac{\partial f_{1}}{\partial y} & \dfrac{\partial f_{1}}{\partial z} \\[1em] \dfrac{\partial f_{2}}{\partial x} & \dfrac{\partial f_{2}}{\partial y} & \dfrac{\partial f_{2}}{\partial z} \\[1em] \dfrac{\partial f_{3}}{\partial x} & \dfrac{\partial f_{3}}{\partial y} & \dfrac{\partial f_{3}}{\partial z} \end{bmatrix}

今、g(t)=(x(t),y(t),z(t))g(t) = \left( x(t), y(t), z(t) \right)としよう。すると

f(x(t),y(t),z(t))=f(g(t))=fg(t) f\left( x(t), y(t), z(t) \right) = f\left( g(t) \right) = f \circ g (t)

f(x(t),y(t),z(t))=f(g(t))=fg(t) \mathbf{f} \left( x(t), y(t), z(t) \right) = \mathbf{f} \left( g(t) \right) = \mathbf{f} \circ g (t)

すると、全微分の連鎖律によって、

dfdt=f(g(t))=f(g(t))g(t)=[fxfyfz][dxdtdydtdzdt]=fxdxdt+fydydt+fzdzdt \dfrac{df}{dt} = f\left( g(t) \right) = f^{\prime}(g(t)) g^{\prime}(t) = \begin{bmatrix} \dfrac{\partial f}{\partial x} & \dfrac{\partial f}{\partial y} & \dfrac{\partial f}{\partial z} \end{bmatrix} \begin{bmatrix} \dfrac{dx}{dt} \\[1em] \dfrac{dy}{dt} \\[1em] \dfrac{dz}{dt} \end{bmatrix} = \dfrac{\partial f}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial f}{\partial y}\dfrac{dy}{dt} + \dfrac{\partial f}{\partial z}\dfrac{dz}{dt}

dfdt=f(g(t))g(t)=[f1xf1yf1zf2xf2yf2zf3xf3yf3z][dxdtdydtdzdt]=[f1xdxdt+f1ydydt+f1zdzdtf2xdxdt+f2ydydt+f2zdzdtf3xdxdt+f3ydydt+f3zdzdt]=[df1dtdf2dtdf3dt] \dfrac{d\mathbf{f}}{dt} = \mathbf{f}^{\prime}(g(t)) g^{\prime}(t) = \begin{bmatrix} \dfrac{\partial f_{1}}{\partial x} & \dfrac{\partial f_{1}}{\partial y} & \dfrac{\partial f_{1}}{\partial z} \\[1em] \dfrac{\partial f_{2}}{\partial x} & \dfrac{\partial f_{2}}{\partial y} & \dfrac{\partial f_{2}}{\partial z} \\[1em] \dfrac{\partial f_{3}}{\partial x} & \dfrac{\partial f_{3}}{\partial y} & \dfrac{\partial f_{3}}{\partial z} \end{bmatrix} \begin{bmatrix} \dfrac{dx}{dt} \\[1em] \dfrac{dy}{dt} \\[1em] \dfrac{dz}{dt} \end{bmatrix} = \begin{bmatrix} \dfrac{\partial f_{1}}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial f_{1}}{\partial y}\dfrac{dy}{dt} + \dfrac{\partial f_{1}}{\partial z}\dfrac{dz}{dt} \\[1em] \dfrac{\partial f_{2}}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial f_{2}}{\partial y}\dfrac{dy}{dt} + \dfrac{\partial f_{2}}{\partial z}\dfrac{dz}{dt} \\[1em] \dfrac{\partial f_{3}}{\partial x}\dfrac{dx}{dt} + \dfrac{\partial f_{3}}{\partial y}\dfrac{dy}{dt} + \dfrac{\partial f_{3}}{\partial z}\dfrac{dz}{dt} \end{bmatrix} = \begin{bmatrix} \dfrac{df_{1}}{dt} \\[1em] \dfrac{df_{2}}{dt} \\[1em] \dfrac{df_{3}}{dt} \end{bmatrix}