I'm beginning Spivak's Calculus on Manifolds and I'm trying to wrap my head around the concept of a derivative as generalized into higher dimensions. I understand that it is a linear transformation, but I have a couple of questions as to how it is a generalization of the one-dimensional case.
If the function goes $\mathbb{R}\rightarrow \mathbb{R}^2$ then it makes sense to speak of the tangent line to the three-dimensional curve drawn by $f$. Given a function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ it makes sense to ask about the inclination of a plane at any point $a\in \mathbb{R}^2$ in which case one could describe the inclination of the plane by means of the partial derivatives with respect to $x$ and $y$, since the inclination in these two directions already fixes the inclination of the plane. I believe this idea can be generalized to describe the $n$th-dimensional space parallel to the function $f:\mathbb{R}^n\rightarrow \mathbb{R}$.
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Can this idea of tangent spaces be extended to functions $\mathbb{R}^n\rightarrow \mathbb{R}^m$?
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And since, in the one-dimensional case, the derivative is often interpreted as describing the inclination of the curve at any point $a$ in the domain, is the derivative in higher dimensions somehow related to the concept of tangent lines, planes, or $n$th-dimensional spaces?
Best Answer
The concept generalizes in a completely straightforward way, with the largest difference mostly being that you can no longer draw the higher dimensional pictures. Let's formalize the geometric tangent plane as follows.
Let $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ be a differentiable function. We consider the graph $\Gamma_f$ of the function as the set of points $(\mathbf{x},f(\mathbf{x})) \in \mathbb{R}^m \times \mathbb{R}^n$. The tangent plane to a point $(\mathbf{x}_0,f(\mathbf{x}_0)) \in \Gamma_f$ is then defined by the set of points $(\mathbf{x},D_{\mathbf{x}_0}f(\mathbf{x}-\mathbf{x}_0) + f(\mathbf{x}_0))$, where $D_{\mathbf{x}_0}f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is the derivative of $f$ at $\mathbf{x}_0$.
For example, given a function $f:\mathbb{R}\rightarrow \mathbb{R}$, this reduces to your regular notion of the tangent line given by $$y = \frac{df}{dx}(x-x_0) + f(x_0).$$ In general, given a function $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$, the tangent plane will be an $m$-dimensional plane which sits within the ambient space $\mathbb{R}^m \times \mathbb{R}^n$. Given a function $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ for example, the tangent plane is a $2$-dimensional plane defined within $\mathbb{R}^2\times \mathbb{R} \cong \mathbb{R}^3$, as you can see through explicit drawings.
All of this is really saying that the derivative is the linear mapping which best approximates a given function in some local neighborhood. The derivative matrix of $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is an $m \times n$ matrix whose $(i,j)$th component tells you how much the tangent plane is inclined in the $j$th direction of the codomain space $\mathbb{R}^n$ when moving in the $i$th direction of the domain space $\mathbb{R}^m$.
Let me finally note that this geometric view of tangent planes is very useful for intuition, but often not so useful in practice. I've been careful to call the objects I've defined tangent planes, because tangent spaces are refer to another concept with precise meaning in differential geometry. Tangent spaces are abstract formalizations of linearizations of smooth manifolds, and derivatives are really linear maps between the tangent spaces of respective manifolds. I think Spivak's book introduces the concept of tangent spaces in the last few chapters.