The matrix $A(u, v)$ displayed in the image is the Jacobian matrix for a vector-valued function $\mathbf{f}(u, v) = (f_1(u, v), f_2(u, v))$, where $f_1$ and $f_2$ are functions of two variables, $u$ and $v$. The Jacobian matrix represents the partial derivatives of each component function with respect to each variable.

Given: $A(u, v) = \begin{pmatrix} \frac{\partial f_1}{\partial u} & \frac{\partial f_1}{\partial v} \\ \frac{\partial f_2}{\partial u} & \frac{\partial f_2}{\partial v} \end{pmatrix}$

To compute this matrix $A(u, v)$, you'll need to know the specific forms of the functions $f_1(u, v)$ and $f_2(u, v)$. Without that information, we can't calculate exact values but can only describe the structure of the matrix.

If you have the functions $f_1$ and $f_2$, please share them, and I can help compute the partial derivatives and form the matrix.

Would you like to proceed with more details on Jacobian matrices, or do you have specific functions for $f_1$ and $f_2$ to evaluate?

Here are some questions to further explore this topic:

1. What are the properties of the Jacobian matrix in a transformation?
2. How do partial derivatives influence the entries of a Jacobian matrix?
3. Can the Jacobian matrix determine whether a transformation is invertible?
4. How does the Jacobian matrix relate to linear approximations of functions?
5. What are some applications of Jacobians in multivariable calculus?

Tip: The determinant of a Jacobian matrix can indicate whether a transformation is locally invertible and, in some contexts, relates to changes in area or volume in transformations.