梯度向量導出之曲線

Example: The temperature at a point (x,y) on a rectangular metal plate is given by T(x,y)=100-2x^2-y^2. Find the path a heat-seeking particle will take, starting at (3,4), as it moves in the direction in which the temperature increases most rapidly.

由梯度向量計算可以得到heat-seeking particle在(x,y)位置時候,找到溫度增加最快的向量為\vec{\nabla}T(x,y)=-4x\hat{i}-2y\hat{j},此梯度向量造成heat-seeking particle位置隨t參數變化,即\frac{d\vec{r}}{dt}=-4x\hat{i}-2y\hat{j},也是熱流的反方向(熱流由高溫位置往低溫位置流動),則可分別在xy分量找關係,即\frac{d\vec{r}}{dt}=\frac{dx}{dt}\hat{i}+\frac{dy}{dt}\hat{j}=-4x\hat{i}-2y\hat{j}

\frac{dx}{dt}=-4xx(t=0)=3,可以解此有起始條件的一階微分方程,得到x(t)=3e^{-4t},另從\frac{dy}{dt}=-2yy(0)=4可找到y(t)=4e^{-2t},則heat-seeking particle的軌跡為\vec{r}(t)=3e^{-4t}\hat{i}+4e^{-2t}\hat{j},下面兩張圖為不同角度下去觀看溫度隨(x,y)座標位置變化、梯度向量及通過(3,4)點的heat-seeking particle路徑,沿著路徑走能到最高溫位置(0,0)

Gradient to Curve
Gradient to Curve

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