A linear approximation of a function $f:\mathbb{R} \to \mathbb{R}$ at the point $x=a$ draws a tangent line to $f(x)$ at $(a,f(a))$ and approximates values of $f(x)$ for $x$ near $a$ by using the function of the tangent line. The same thing is done for functions $f: \mathbb{R}^2 \to \mathbb{R}$, but with a tangent plane instead of a line. Here's my question:
I'm having trouble understanding what the linear approximation for a vector valued function $\vec{F}:D \subset \mathbb{R}^n \to \mathbb{R}^m$, which has the form $\vec{F}(\vec{x}) \approx \vec{F}(\vec{a}) + A(\vec{a})(\vec{x}-\vec{a})$ where $A_{ij}(\vec{a}) = \displaystyle{\frac{\partial F_i}{\partial x_j}(\vec{a})}$, repersents graphically. Any help is appreciated, thanks in advance 🙂
P.S. if you want to demonstrate it with an example, that's fine. Just please use a function in $\mathbb{R}^3$.
Best Answer
To begin with, I would like to point out that the graphical interpretations of functions $f_1:\mathbb R \to \mathbb R$ and $f_2:\mathbb R^2 \to \mathbb R$ are just interpretations of the function using mathematical objects that never appear in the actual functions themselves.
For $f_1:\mathbb R \to \mathbb R$ we plot points in a Cartesian plane whose coordinates are in $\mathbb R^2.$ Where do you see $\mathbb R^2$ in the definition of $f_1$? It is nowhere in there. But by taking the Cartesian product of the domain and range we get $\mathbb R \times \mathbb R \sim \mathbb R^2,$ which we use as coordinates of a two-dimensional plane. The set $\{(x,y) \in \mathbb R^2 \mid f_1(x) = y \}$ is a figure in that plane; we call that figure the graph of $f_1,$ and if $f_1$ is differentiable at $x_0$ then part of that graph is a curve through the point with coordinates $(x_0,f(x_0))$ that has a tangent line at that point.
This is a very useful visualization for a real-valued function on the real numbers, but let's just be clear that it is something we invented by pulling numbers out of the function definition and recombining them in a way that the function definition never said we should do. (It also never said we could not do this, and it turns out that it works, so we do it.)
Likewise for $f_2:\mathbb R^2 \to \mathbb R$ we take each function value $z = f_2(x,y)$ and plot it at the point $(x,y,z)$ in a three-dimensional Cartesian space with coordinates $\mathbb R^3,$ even though there is no three-dimensional space or the set $\mathbb R^3$ anywhere in the definition of $f_2.$
In short, we took the coordinates of the function's input and concatenated the function's output as a third coordinate in order to produce a graph.
You can do the same thing with $\vec F:D\subset\mathbb R^n \to \mathbb R^m.$ Take the $n$ coordinates of each $\vec x\in D$ and concatenate the $m$ coordinates of $\vec F(\vec x)$ to produce the coordinates of a point in an $(n+m)$-dimensional Cartesian space, $\mathbb R^{n+m}.$ The derivative of $\vec F$ at some input value $\vec x$ then corresponds to an $n$-dimensional hyperplane in that space.
This does not help much with visualization, however, since visualizing anything in more than three dimensions is very hard for most people.