Transcription
A simple function of one variable where f of x equals x squared. When we take the derivative of this function, we get df over dx equals 2x, which represents how the function changes as we vary the input x.
Moving forward into the realm of multivariable calculus, we encounter functions that can accept multiple inputs simultaneously, such as a function f that depends on both x1 and x2. In this new territory, we discover partial derivatives, which tell us how the function changes with respect to each individual variable while keeping all other variables constant. We compute the partial derivative of f with respect to x1 and separately the partial derivative of f with respect to x2.
Now comes the beautiful leap that defines the Jacobian matrix. What happens when we have not just one function of multiple variables, but multiple functions, each depending on multiple variables? Imagine a vector-valued function f that takes a vector containing x1 and x2 as input and produces a vector containing two output functions, f1 of x1 and x2 and f2 of x1 and x2. This scenario gives us a mapping from a two-dimensional input space to a two-dimensional output space.
The Jacobian matrix elegantly captures all possible derivative relationships in this multi-function multivariable system by organizing them into a systematic grid. Since we have two functions and two variables, we get exactly four partial derivatives. The partial of the first function with respect to the first variable, the partial of the first function with respect to the second variable, the partial of the second function with respect to the first variable, and the partial of the second function with respect to the second variable. These four derivatives form a 2x2 matrix that we call the Jacobian matrix, denoted as J subscript f, which contains all the local linear approximation information about how our vector-valued function behaves.