Dimensional analysis

Mathematical statement

Relative to basic dimensions L,M,T, write [Q]=L^a M^b T^c, represented by (a,b,c). Products add vectors, quotients subtract them and integer powers scale them. Quantities to be equated or added must have matching dimensions.

Representative formula (exact TeX)

d(Q)

Assumptions

• Exercises use only the three basic dimensions L,M,T and integer exponents.
• Dimensional consistency is necessary but not sufficient for a physical equation; dimensionless and dimensioned constants must be explicitly specified.

Symbol scope

• Q — physical quantity
• L — length dimension
• M — mass dimension
• T — time dimension

Intuition, definition, and misconceptions

Velocity’s dimension comes from length divided by time. Products and quotients combine exponents separately for each basic dimension.

A pure numerical constant cannot fix inconsistent dimensions. Consistent dimensions may still accompany an incorrect coefficient, direction or physical assumption.

Dimensional analysis does not track numerical unit-conversion factors and does not replace experiments, boundary conditions or model validation.

Do not treat dimensional consistency as proof of an equation or identify unit names completely with dimension exponents.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Ratio and proportion — Division and dimensionless ratios use ratio semantics.
• Arithmetic operations — Exponent vectors use addition, subtraction and integer scaling.
• Equation — Compare the dimensions of both equation sides.

Verification boundary

This slice checks only L,M,T exponent vectors; consistency is not promoted to physical correctness.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add SI unit conversions, Buckingham Pi and nondimensionalization.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.