In this lecture, we'll cover the following:

• Applying the chain rule to relate derivatives of shape functions in natural and physical coordinates.
• Defining and interpreting the Jacobian matrix as a coordinate mapping.
• Constructing and inverting the Jacobian matrix.
• Using the inverse Jacobian to compute derivatives needed for the strain–displacement matrix $B$.
• Calculating Jacobian components from nodal coordinates and shape function derivatives.

In this lecture, we focus on how to compute the derivatives of shape functions with respect to physical coordinates by applying the chain rule, since the shape functions are originally defined in natural coordinates. We show how this leads naturally to the definition of the Jacobian matrix, which acts as a mapping between derivatives in the natural coordinate system and those in the physical coordinate system. We then express this relationship in matrix form and explain why the inverse of the Jacobian is required to obtain the derivatives needed for constructing the strain–displacement matrix $B$.

We also work through how to compute the individual components of the Jacobian matrix using nodal coordinates and derivatives of the shape functions. Once these components are assembled into a $2\times 2$ matrix, we invert it and use it to transform derivatives from natural to physical coordinates. This process is presented as a key step in forming the strain–displacement matrix, highlighting the central role of the Jacobian in finite element formulations.

Next up

Before moving on to implementation, the next lecture pauses to recap and consolidate all the theoretical components developed so far.