Consider the following homogeneous deformation of a continuum body at an
instant in time:
$\mathbf{x} = \mathbf{x}(\mathbf{X}, t) = 3 X_1 \mathbf{E}_1 + 2 X_2 \mathbf{E}_2 + 5 X_3 \mathbf{E}_3$
where the bases $\{\mathbf{E}_A\}$ and $\{\mathbf{e}_i\}$ are perfectly aligned.
a) Obtain the equation for a surface, S, occupied in the current (deformed)
configuration by the material lying on a sphere, defined by $X_1^2 + X_2^2 + X_3^2 = 1$ in
the reference configuration. (I encourage you to sketch the surfaces in the reference
and current configurations)
b) Calculate the volume enclosed by S in the current configuration.
c) Calculate the change in the infinitesimal area elements of the sphere
$X_1^2 + X_2^2 + X_3^2 = 1$ at three locations, where the outward normal to the surface at
those locations is $\mathbf{N} = \mathbf{E}_1$, $\mathbf{N} = \mathbf{E}_2$, $\mathbf{N} = \mathbf{E}_3$.
