Total differential and linearization

Level UniversityDifficulty ★★★★★Concept⌖ Open in the map

What is it?

Near a point, a differentiable function is approximately linear: $\dd f = \sum_i \frac{\partial f}{\partial x_i}\dd x_i$ . The graph has a tangent plane, and small input errors propagate linearly.

Formulas

\dd f = \frac{\partial f}{\partial x}\dd x + \frac{\partial f}{\partial y}\dd y, \qquad z = f(a) + \nabla f(a)\cdot(x - a)

Where it shows up in computing

Kalman filter★★★★★frequentRobotics and control
The extended Kalman filter linearizes nonlinear dynamics and sensors at the current estimate.
Control theory★★★★★frequentRobotics and control
Linearization around an equilibrium gives the matrices $A$ , $B$ of a state-space model.

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚙ Robotics and control

Control theory★★★★★
Kalman filter★★★★★
Control theory→Trajectory optimization and MPC★★★★★

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.