Linearization in Several Variables

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

1. Choose a nearby base point with easy exact values.
2. Compute $f$ and every first partial derivative there.
3. Write the full dot-product or coordinate linearization.
4. Insert signed coordinate increments.
5. Use curvature, scale, or a numerical check to assess the approximation.

Fully worked example

Graphical or geometric meaning

The tangent plane touches the cone-shaped distance surface at $(3,4,5)$. The displacement has one positive and one negative coordinate component, and their gradient-weighted contributions partly cancel.

Common mistakes and why they fail

Verification and reasonableness checks

• Verify $L(\mathbf a)=f(\mathbf a)$.
• Confirm partial derivatives of $L$ match those of $f$ at the base.
• Compare with a calculator value for a nearby point.

The tangent plane predicts coupled changes

For differentiable $f(x,y)$ near $(a,b)$, $L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)$. Both input changes contribute, and the coefficients must be evaluated at the same base point. Differentiability, stronger than merely having partial derivatives, guarantees the error is small compared with the distance to the base point. Write $\Delta x$ and $\Delta y$ with signs visible and check units in each term. The gradient supplies the slope coefficients, so the approximation agrees with every directional derivative at the point. Curvature and distance control accuracy; large motion in either coordinate can make a tangent-plane estimate poor even if the other change is tiny.

Practice

1. Linearize $x^2+y^2$ at $(1,1)$.
2. Approximate $1.01^2+0.98^2$ from $(1,1)$.
3. What is the output dimension of a scalar linearization?

Connections and next steps

• Tangent Planes
• Gradient
• Differentials