Surface z = f(x, y) = 4 − 0.3x² − 0.2y² with point of tangency P = (1, 1, 3.5). Step away from P by Δx and Δy, and compare the rise on the tangent plane with the rise on the surface.
Drag to rotate
surface
tangent plane
slice y = b, slope fₓ
slice x = a, slope fy
error Δz − dz
L(x, y) = f(a, b) + fₓ(a, b)(x − a) + fy(a, b)(y − b)
L(x, y) = 3.5 − 0.6(x − 1) − 0.4(y − 1)
| fₓ(1, 1) | |
| fy(1, 1) | |
| dz = fₓΔx + fyΔy | |
| Δz = f(a+Δx, b+Δy) − f(a, b) | |
| Error |Δz − dz| |
The orange and green curves are the two vertical slices through P. Their tangent lines (dashed) have slopes fₓ and fy, and together they determine the tangent plane.
The thick orange and green steps show the linearization as a walk: move Δx and rise fₓΔx, then move Δy and rise fyΔy. The total rise on the plane is dz.
The red gap between the plane and the surface is the approximation error. Make Δx and Δy small and the error shrinks much faster than the step itself. That is what locally linear means.