# Tangent mapping

These images show the initial configuration of a body (square) and a nonlinear deformation of that body into a curvy shape (to the right of the square).  Overlaid on the actual deformed shape is the so-called tangent mapping at the indicated point.  It coincides with the nonlinear mapping to first-order accuracy.

### Background

Grade school children (or college freshman in some countries) learn that any nonlinear function $y=f(x)$ has a local tangent mapping at any point of interest, $x_0$, which is simply the straight-line tangent to the nonlinear curve at that point.  The equation of the tangent mapping in this case is

$y-y_0 = m (x-x_0)$, where $m=dy/dx$ evaluated at $x=x_0$ and $y_0=f(x_0)$.

Analogously, consider the nonlinear “material mapping” transformation commonly studied in continuum mechanics:

Input:  Initial position vector of a point $\mathbf{X}$ in the body

Output: Deformed position vector $\mathbf{x}$ of the same point.

The “output” (deformed location of a point)  is generally nonlinearly related to the “input” (initial location of a point), which is why the initial square in the animations deforms to a curvy shape. However, just as the nonlinear function $y=f(x)$ has a local straight-line tangent line, this vector-to-vector transform also has a tangent mapping.  The tangent mapping at a point $\mathbf{X}_0$ in the initial configuration of the body (red dot in the animations) is defined by

$\mathbf{x}-\mathbf{x}_0 = \mathbf{F}\cdot(\mathbf{X}-\mathbf{X}_0)$,

where

$\mathbf{F}$ is the deformation gradient tensor having components $F_{ij}=\partial{x_i}/\partial{X_j}$

$\mathbf{x}_0$ is the deformed location of the point initially located at $\mathbf{X}_0$ (blue dot in the animations)

Evidently, the deformation gradient tensor plays a role similar to the role played by slope $m=dy/dx$ in the scalar case of $y=f(x)$.