∇
VECTOR Calc 1–4
4d Streak
120 XP
Mode A: Guided Applet
x₀ = 1.00| f(x₀) = 0.750| f'(x₀) = 0.250
Dynamic Parameters
Target Point x₀ (x0x_0)1.0
Epsilon Corridor ε (ε\varepsilon)0.50
Delta Window δ (δ\delta)0.35
Visual Overlays & Layers
Chapter 4 Theory

Directional Derivatives, The Gradient Vector & Tangent Planes

Steepest ascent vector ∇f, orthogonal level curves, and local linearization

Notice how the gradient vector ∇f points directly uphill in the direction of greatest slope, while remaining strictly perpendicular to the level contour lines.

1. The Steepest Ascent Property of ∇f

The directional derivative D_u f = ∇f · u reaches its maximum value |∇f| when unit vector u points in the exact direction of ∇f (cos θ = 1). Conversely, moving along a contour line has zero change (∇f · u = 0), proving ∇f is orthogonal to level curves.

∇f(x0,y0)⊥{(x,y):f(x,y)=c}\nabla f(x_0, y_0) \perp \{ (x,y) : f(x,y) = c \}

Quick Reference & Cheat Sheet

Tangent Plane Equation
z−z0=fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)
The unique flat plane whose directional slopes match the surface in all 360° directions.