Problem 7.7

We seek the jacobian (\(\frac{\partial x}{\partial \xi}\)) of the the spatial mapping

\[\begin{equation*} x(\xi) = \sum_{a=1}^2 x_{A+a-1} N_a ( \xi) \end{equation*}\]

with linear shape functions

\[\begin{equation*} N_a(\xi) = \frac{1}{2} \left[ 1 + (-1)^a \xi \right]. \end{equation*}\]

First, let us take the derivative of the mapping with respect to \(\xi\).

\[\begin{equation*} \frac{d x}{d\xi} = \sum_{a=1}^2 x_{A+a-1} \frac{d}{d\xi} N_a ( \xi) \quad. \end{equation*}\]

We therefore need the derivatives of the shape functions:

\[\begin{equation*} \frac{d}{d\xi} N_a(\xi) = \frac{1}{2} (-1)^a \end{equation*}\]

such that the derivatives of the two shape functions are

\[\begin{align*} \frac{d N_1}{d\xi} = -\frac{1}{2} \\ \frac{d N_2}{d\xi} = +\frac{1}{2} \end{align*}\]

We may then write the summation in the jacobian explicitly as

\[\begin{align*} \frac{d x}{d\xi} &= x_{A} \frac{d N_1}{d\xi} + x_{A+1} \frac{d N_2}{d\xi} \\ &= \frac{1}{2} \left[ x_{A+1} - x_{A} \right] \\ &= \frac{1}{2} \Delta x_{A} \end{align*}\]

as required.