Field Derivatives: The Jacobian — Lesson 6

In this lesson, we expand further on the finite-dimensional weak form, over which we have the partitions into elements of domains. Then using the elements whose surfaces coincide with the Neumann boundary, we will see how to compute the gradients of basis functions. Lastly, we will discuss how to parametrize them in terms of the coordinates in the parent subdomain while developing the basis functions.

In this video, we recall the mapping and write it using coordinate notation. Then, as we did in the 1D problem, we will compute the derivative. By recalling the chain rule, we will see how to get the inverse of that to get the derivative. For this, we will use a point-to-point vector map and compute the gradient of the map, or a tangent map. The tangent map is denoted as a tensor J - also known as the Jacobian of the map. Later, we will represent the Jacobian as a matrix and then as an inverted matrix, which we need for our chain rule.