(Solution) - If the temperature of a plate at the point x -(2025 Original AI-Free Solution)

Paper Details

If the temperature of a plate at the point (x, y) is T(x, y) = 10 + x2 - y2, find the path a heat-seeking particle (which always moves in the direction of greatest increase in temperature) would follow if it starts at (-2, 1). The particle moves in the direction of the gradient
(T = 2xi - 2yj
We may write the path in parametric form as
r(t) = x(t)i + y(t)j
And we want x(0) = -2 and y(0) = 1. To move in the required direction means that r'(t) should be parallel to VT. This will be satisfied if