Linear Approximations

Suppose pollution concentration across a city is modeled as a differentiable function of x and y and represented by a 3D surface above the xy-plane. Because the concentration changes smoothly, a nearby value can be estimated from a known measurement.

This process is called linear approximation. It uses a local flat model to estimate the pollution level near a specific point, such as a monitoring station where the concentration is known.

While the pollution surface may curve across the city, it acts like a flat sheet very close to that station. This sheet is called the tangent plane, defined by the known concentration and the partial derivatives in the x and y directions.

So, the concentration at a nearby location is estimated using the tangent plane rather than the curved surface. When the location is very close to the monitoring station, the estimate is usually accurate.

But as the location moves farther away, the approximation becomes less accurate because the pollution surface no longer stays close to that flat model. In this way, linear approximation gives a quick local estimate of a complex surface.