Editing Infinitesimal strain theory (section)

===Physical interpretation===
From [[Finite deformation tensor|finite strain theory]] we have
<math display="block">d\mathbf{x}^2 - d\mathbf{X}^2 = d\mathbf X \cdot 2\mathbf E \cdot d\mathbf X \quad\text{or}\quad (dx)^2 - (dX)^2 = 2E_{KL}\,dX_K\,dX_L</math>

For infinitesimal strains then we have
<math display="block">d\mathbf{x}^2 - d\mathbf{X}^2 = d\mathbf X \cdot 2\mathbf{\boldsymbol \varepsilon} \cdot d\mathbf X \quad\text{or}\quad (dx)^2 - (dX)^2 = 2\varepsilon_{KL}\,dX_K\,dX_L</math>

Dividing by <math>(dX)^2</math> we have
<math display="block">\frac{dx-dX}{dX}\frac{dx+dX}{dX}=2\varepsilon_{ij}\frac{dX_i}{dX}\frac{dX_j}{dX}</math>

For small deformations we assume that <math>dx \approx dX</math>, thus the second term of the left hand side becomes: <math>\frac{dx+dX}{dX} \approx 2</math>.

Then we have
<math display="block">\frac{dx-dX}{dX} = \varepsilon_{ij}N_iN_j = \mathbf N \cdot \boldsymbol \varepsilon \cdot \mathbf N</math>
where <math>N_i=\frac{dX_i}{dX}</math>, is the unit vector in the direction of <math>d\mathbf X</math>, and the left-hand-side expression is the [[Deformation (mechanics)#Strain measures|normal strain]] <math>e_{(\mathbf N)}</math> in the direction of <math>\mathbf N</math>. For the particular case of <math>\mathbf N</math> in the <math>X_1</math> direction, i.e., <math>\mathbf N = \mathbf I_1</math>, we have
<math display="block">e_{(\mathbf I_1)}=\mathbf I_1 \cdot \boldsymbol \varepsilon \cdot \mathbf I_1 = \varepsilon_{11}.</math>

Similarly, for <math>\mathbf N=\mathbf I_2</math> and <math>\mathbf N=\mathbf I_3</math> we can find the normal strains <math>\varepsilon_{22}</math> and <math>\varepsilon_{33}</math>, respectively. Therefore, the diagonal elements of the infinitesimal strain tensor are the normal strains in the coordinate directions.