Gradient and directional change

The gradient points uphill fastest

The gradient collects directional slopes into a vector. Its direction is the direction of greatest increase of a scalar field, and its magnitude is the largest increase per unit distance.

Use the directional derivative

For a unit direction n̂, the directional derivative is D_nφ = ∇φ·n̂. By the dot-product bound, this is largest when n̂ points along ∇φ and equals |∇φ|; it is zero along a level surface because ∇φ is perpendicular to contours.

For φ = 4x − 3y, ∇φ = (4,−3) and |∇φ| = 5. Moving one metre along n̂ = (4,−3)/5 raises φ by 5 units; moving perpendicular to it produces no first-order change.

Gradient signs depend on the field being measured

For temperature, heat flux points opposite ∇T because heat flows downhill: q = −k∇T. For pressure-driven flow, a pressure gradient can drive motion in the opposite direction.

Work it: For h(x,y) = 100 − x² − 2y², find the steepest ascent direction and rate at (1,2). Predict whether moving east raises or lowers h.