The concept of dimension is important because every mathematical equation linking physical quantities must be dimensionally consistent, implying that mathematical equations must meet the following two rules. The first rule is that, in an equation, the expressions on each side of the equal sign must have the same dimensions. This is fairly intuitive since we can only add or subtract quantities of the same type (dimension). The second rule states that, in an equation, the arguments of any of the standard mathematical functions like trigonometric functions, logarithms, or exponential functions must be dimensionless.
If either of these two rules is violated, the equation is dimensionally inconsistent, hence it cannot be a representation of the correct statement of any physical law. Dimensional analysis can check for mistakes or typos in algebra, help remember the various laws of physics, and even suggest the form that new laws of physics might take.
Let us understand the effect of the operations of calculus on dimensions. The derivative of a function is the slope of the line tangent to its graph, and slopes are ratios. Thus, for physical quantities, say v and t, the dimension of the derivative of v with respect to t is the ratio of the dimension of v over that of t. Similarly, since integrals are just sums of products, the dimension of the integral of v with respect to t is simply the dimension of v times the dimension of t.
This text is adapted from Openstax, University Physics Volume 1, Section 1.4: Dimensional Analysis.
