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Section 23.3 Differentiability in higher dimensions
Objectives
Understand the concept of differentiability in higher dimensions and how it differs from single-variable differentiability.
Apply the definition of differentiability to compute derivatives of multivariable functions.
Use the Jacobian matrix to analyze the behavior of multivariable functions and their derivatives.
A function \(f:\bb R^n\to\bb R^m\) is differentiable at a point \(a\in\bb R^n\) if there exists a linear transformation \(L:\bb R^n\to\bb R^m\) such that
\begin{equation*}
\lim_{h\to 0}\frac{\norm{f(a+h)-f(a)-L(h)}}{\norm{h}}=0\text{.}
\end{equation*}
In this case, \(L\) is called the derivative of \(f\) at \(a\text{,}\) denoted by \(D f(a)\text{.}\)
The existence of partial derivatives does not guarantee differentiability. For example, the function
\begin{equation*}
f(x,y) = \begin{cases}
\dfrac{y^3}{x^2+y^2} \amp (x,y)\neq(0,0) \\
0 \amp (x,y)=(0,0)
\end{cases}
\end{equation*}
is continuous and has partial derivatives at the origin, but it is not differentiable there.
If
\(f:\bb R^n\to\bb R^m\) is differentiable at
\(a\in\bb R^n\text{,}\) then the derivative
\(D f(a)\) can be represented as a matrix, which just ends up being the Jacobian matrix.
