Physics I / The Chain Rule and Directional Derivatives
Practice question · Multiple choice

A bug walks across a heated metal plate. Its GPS reports position (x(t),y(t))(x(t), y(t)); the plate’s temperature map is T(x,y)T(x,y). What computes the warming rate the bug feels, dT/dtdT/dt?

Hints
  1. The bug feels change from moving in xx and from moving in yy; each contribution is (sensitivity) × (rate of motion).
  2. This weighted sum is the multivariable chain rule, equivalently ∇T⋅v⃗\nabla T \cdot \vec{v}.
Show the answer

A. Tx x′+Ty y′T_x\,x' + T_y\,y', each partial times its speed

Why

The chain rule composes the map’s sensitivities (Tx,TyT_x, T_y) with the bug’s velocity: dT/dt=∇T⋅v⃗dT/dt = \nabla T\cdot\vec{v}. Walk along an isotherm and it is zero; walk up the gradient and it is maximal, the dot product does the accounting.

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