Systems of differential equations: Non-linear differential equations
Jacobian matrix
For real functions of several variables, say \(f:\mathbb{R}^n\longrightarrow \mathbb{R}\), where is \(n\) a natural number, we have already introduced partial derivatives and gradient \(\nabla f\) in the chapter Functions of several variables; in row vector notation this looks like: \[\nabla f= \matrix{\dfrac{\partial f}{\partial x_1} & \dfrac{\partial f}{\partial x_2} & \ldots & \dfrac{\partial f}{\partial x_n}}\] We are now going a step further and consider a vector-valued function \(f = (f_1,\ldots, f_k):\mathbb{R}^n\longrightarrow \mathbb{R}^k\) where \(k\) and \(n\) are natural numbers. Now we can introduce the Jacobian matrix,
The Jacobian matrix \(J(f)\) is defined as the \(k\times n\) matrix of all possible partial derivatives: \[J(f)= \matrix{\dfrac{\partial f_1}{\partial x_1} & \ldots & \dfrac{\partial f_1}{\partial x_n}\\ \vdots & \ddots & \vdots\\ \dfrac{\partial f_k}{\partial x_1} & \ldots & \dfrac{\partial f_k}{\partial x_n}}\]
Linearization We can use the Jacobian matrix approximate an arbitrary vector-valued function as a linear mapping: \[\cv{f_1(x_1,\ldots,x_n)\\ \vdots \\ f_k(x_1,\ldots,x_n)}\approx \cv{f_1(a_1,\ldots,a_n)\\ \vdots \\ f_k(a_1,\ldots,a_n)} + J(f)(a_1,,\ldots,a_n)\cdot \cv{x_1-a_1\\ \vdots \\ x_n-a_n}\]
Consider the following vector-valued function: \[f(x,y)=\bigl(f(x,y), g(x,y)\bigr)=\bigl(\sin(x+2y), 3e^x+5y\bigr)\] Then: \[\begin{aligned} f(x,y) &\approx f(0,0)+\matrix{\dfrac{\partial f}{\partial x}(0,0) & \dfrac{\partial f}{\partial y}(0,0)\\ \dfrac{\partial f}{\partial x}(0,0) & \dfrac{\partial f}{\partial y}(0,0)}\cdot \cv{x\\y}\\ \\ &= \cv{0\\ 3}+\matrix{1 & 2\\3 & 5}\cv{x\\y} \\ \\ &= \cv{x+2y\\ 3 + 3x+ 5y}\end{aligned}\]
