Differentials

Differentials is a term used in several fields to denote changes or gear systems that accommodate uneven speeds. In mathematics and physics, a differential describes an infinitesimal change in a quantity, while in engineering and mechanics it refers to a gear train that distributes power while allowing components to rotate at different speeds.

In calculus, if y = f(x), the differential dy represents the approximate change in y for a small change dx, given by dy = f'(x) dx. For a function of several variables z = f(x1, x2, ..., xn), the total differential is dz = ∑ ∂f/∂xi dxi. Differentials underpin linear approximation and, in differential geometry, form the basis of differential forms where dx^i are local one-forms and the exterior derivative d acts to produce higher-order forms. Differentials obey rules such as linearity and the product rule d(uv) = u dv + v du, and they can be used in change-of-variables and integration methods within the broader framework of calculus.

In mechanics, a differential is a gear arrangement that splits input power to two or more outputs while allowing differing output speeds. The most common form is the open differential, which enables wheels on an axle to rotate at different speeds, improving turning manners on low-slip conditions. Variants include limited-slip differentials, which reduce wheel spin under load, and locking differentials, which rigidly couple wheel speeds for traction in challenging conditions. Automotive differentiials are essential for safe handling and efficient power transfer in vehicles.