題組內容

1. A heat-seeking particle is located at the point (-1,2) on a metal plate whose temperature at (x, y) is T(u,u) = 64-2x2 y2.

(b) (6 分) Find the path of the particle as it continuously moves in the direction of maximum temperature increase.

詳解 (共 1 筆)

助人為本
助人為本
詳解 #7416552
2026/06/22
所以看到moves in the direction of maximum temperature increase
用梯度來解
前面那一題有算出梯度為(-4xy^2,-4x^2y)
所以求路徑斜率我們dy/dx
6a391c4db7ca8.jpg
所以交叉相乘可以得到ydy=xdx
得到y^2-x^2=C
然後代入(-1,2)
可以得到C=3
所以可以得到路徑方程式y^2-x^2=3
所以最大增加方向不是隨機的，而是沿著梯度走的方向