Total Differential & Error Propagation

The total differential estimates how small changes in inputs change the output. Learn df = f_x dx + f_y dy, linear approximation of surfaces, absolute and relative error propagation, worst-case vs statistical (root-sum-square) combinations, and sensitivity coefficients.

Every measurement has uncertainty. If a resistor and a current are each known to ±1 %, how accurate is the computed power? The total differential answers this and shows which input dominates the error.

An analogy

The total differential is like a recipe tolerance: small errors in each ingredient add up in the final dish, weighted by how sensitive the dish is to each one.

Linear approximation

Near a point, a smooth surface looks like its tangent plane: f(x + dx, y + dy) ≈ f + f_x dx + f_y dy.

Worst-case vs statistical

Worst-case adds absolute contributions (conservative, for safety-critical tolerances). If errors are independent and random, RSS gives a more realistic estimate.

Relative errors for products and powers

For products and quotients, relative errors add (weighted by exponents): V = πr²h → ΔV/V ≈ 2Δr/r + Δh/h. Squared quantities double their relative error.

Sensitivity analysis

Normalised sensitivities (∂f/∂x)(x/f) rank which parameters matter; design effort goes to tightening the dominant ones. Monte Carlo simulation handles non-linear or large errors.

Formulas

• Total differential: df ≈ (∂f/∂x)dx + (∂f/∂y)dy
• Worst case: |Δf| ≤ |f_x|Δx + |f_y|Δy
• Root-sum-square (independent): Δf = √((f_x Δx)² + (f_y Δy)²)
• Power laws: f = k xᵃ yᵇ ⇒ Δf/f ≈ |a|Δx/x + |b|Δy/y

Worked example: Power uncertainty

Given: I = 2 ± 0.02 A, R = 10 ± 0.1 Ω

Find: ΔP.

• P = 40 W
• dP = 2IR dI + I² dR = 0.8 + 0.4 = 1.2 W
• RSS: √(0.8² + 0.4²) = 0.89 W

Answer: 40 ± 1.2 W worst case (± 0.9 W RSS)

Worked example: Cylinder volume

Given: r = 5 ± 0.05 cm, h = 10 ± 0.1 cm

Find: ΔV/V.

• 2 × 1 % + 1 % = 3 %

Answer: ± 3 % (V = 785 ± 24 cm³)

Common mistakes

• Wrong: Adding absolute errors of different units. — Right: Weight by sensitivities first.
• Wrong: Using RSS for correlated errors. — Right: RSS assumes independence.

Applications

• Lab measurements: Uncertainty budgets for experiments.
• Manufacturing tolerances: Stack-up analysis.
• Sensor design: Which component tolerance dominates.

Key takeaways

• df = Σ (∂f/∂xᵢ) dxᵢ.
• Worst case adds magnitudes; RSS for independent errors.
• Relative errors add weighted by exponents.
• Find and tighten the dominant contributor.

Going further

• GUM: Guide to the Expression of Uncertainty in Measurement.
• Monte Carlo propagation: Non-linear uncertainty analysis.
• Covariance terms: Correlated input errors.