Local Linearization

One main theme in calculus is that "sufficiently small changes in smooth functions are nearly linear, even on the scale of the input change."  Analytically this is an approximation of the change in a function by its total differential,

where  as  and .  (Note:  dx and dy are just translated local variables or differences from a fixed point of tangency .)  If we divide both sides of this equation by the size of the vector change, , the remaining error term   is still small for sufficiently small change .

Geometrically, this division means that a sufficiently magnified view of the smoothly curved graph   appears to be the same as its flat tangent given by  in translated local coordinates, where   and  with   fixed.  Division by   magnifies so that the input change appears to be a unit vector.  This magnification is illustrated in the closed cells below for explicit, implicit, and parametric graphs, as well as for vector fields.

Practically, this approximation lets us replace the study of a nonlinear expression   with a linear expression   for "sufficiently small" changes.  (Technically we want "smooth = " or   locally uniformly, so that we can "move the microscope" a small amount without changing the power.  This is equivalent to having continuous partial derivatives.  Pointwise convergence is sufficient for tangency at one point, but the Implicit Function Theorem and several integration theorems need full smoothness.)

The translated dx-dy-dz-coordinate system is helpful in the magnified graphs below because it stays at the center of the microscopic image, while the original x-y-z-origin moves farther away as the magnification increases.

An explicit graph and tangent

A contour (implicit) graph and tangent

A parametric curve graph and tangent

A parametric surface graph and tangent

Local Variables and a Common Procedure to Find Tangents

Linearization of vector fields and the Theorems of Green, Gauss, & Stokes