Section 4 Linear algebra in multivariable calculus
Subsection 4.1 Differentiability
A function \(f\colon \R^n\to \R^m\) is differentiable at the point \(\mathbf{x}_0\) if there exists a linear transformation \(L\colon \R^n\to \R^m\) such that
If \(L\) exists, it is called the derivative of \(f\) at \(\mathbf{x}_0\) denoted \(Df(\mathbf{x_0})\).
To understand the definition of the derivative, start with the case \(n=m=1\text{.}\) The derivative of \(f\) at \(x_0\) is a number \(f'(x_0)\) such that
for \(h\) near \(0\text{.}\) The meaning of "approximately equals \(\ldots\) for \(h\) near \(0\)" is made precise by using a limit. To generalize to higher dimensions, interpret \(f'(x_0)h\) as the value of a linear transformation that sends \(h\) to \(f'(x_0)h\text{.}\) The derivative \(Df(\mathbf{x}_0)\) of \(f\) at \(\mathbf{x_0}\) is a linear transformation such that
for \(\mathbf{h}\) near \(\mathbf{0}\text{.}\) Putting \(\mathbf{h} = t\mathbf{e}_j\text{,}\) this reads
for \(t\) near \(0\text{.}\) Dividing both sides by \(t\) and taking a limit, we get an expression for \(Df(\mathbf{x}_0)\mathbf{e}_j\text{.}\)
where \(\mathbf{y}=(y_1,y_2,\ldots,y_m) = f(x_1,x_2,\ldots,x_n) = f(\mathbf{x})\text{.}\) From this it follows that \(Df(\mathbf{x}_0)\text{,}\) if it exists, is represented by the matrix \(\left[\frac{\partial y_i}{\partial x_j}\right]\text{.}\)
Subsection 4.2 The Chain Rule
Consider the composition of functions
and suppose \(g\) is differentiable at \(\mathbf{t}_0\) and \(f\) is differentiable at \(\mathbf{x}_0=g(\mathbf{t}_0)\text{.}\) The chain rule says that \(f\circ g\) is differentiable at \(\mathbf{t}_0\text{,}\) and that the derivative of the composition is the composition of the derivatives.
This explains the "tree diagram rule" given in most multivariate calculus texts. The partial derivative \(\frac{\partial y_i}{\partial t_j}\) is just the \(i,j\) entry of the product of the derivative matrices for \(f\) and \(g\text{.}\)
Exercises 4.3 Exercises
1.
Verify the definition of differentiable function (4.1) given in is equivalent to the usual definition for \(n=m=1\) from Calculus 1.
2.
Explain equation (4.2). Why does the limit on the left equal the vector on the right?
3.
Explain equation (4.3). How does this equation follow from the previous?
4.
Explain equation (4.5). How is it the same as the chain rule (4.4)?