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Math101Linearization in Several Variables
A rigorous, example-driven guide to linearization in several variables, including hypotheses, method choice, verification, and practice.
The central idea
If $f:\mathbb R^n\to\mathbb R$ is differentiable at $\mathbf a$, then $L(\mathbf x)=f(\mathbf a)+\nabla f(\mathbf a)\cdot(\mathbf x-\mathbf a)$ is its linearization. For $f(x,y)$, this is the tangent-plane function. Differentiability means the error is small relative to $\|\mathbf x-\mathbf a\|$.
Definitions, hypotheses, and notation
Total differential notation writes $df=f_xdx+f_ydy$ at the base point. It is the change predicted by the linearization, while the exact finite change $\Delta f$ includes higher-order effects. If inputs carry measurement errors, $|f_x||dx|+|f_y||dy|$ gives a conservative first-order absolute error estimate.
The base point should be close in the same coordinate scaling used by the model. A function can be differentiable while curvature makes a linearization poor far away. Small coordinate increments, consistent units, and an independent magnitude check keep the local approximation honest.
Conceptual meaning
A differentiable surface looks planar under sufficient magnification. The gradient supplies the slope in every coordinate direction at once, so one linear map predicts changes from any small input displacement.
A dependable method and decision rule
- Choose a nearby base point with easy exact values.
- Compute $f$ and every first partial derivative there.
- Write the full dot-product or coordinate linearization.
- Insert signed coordinate increments.
- Use curvature, scale, or a numerical check to assess the approximation.
