11 Partial derivatives, part 3

Exercise 1: Identifying coordinate functions

Each of the following functions has more than one output. Identify the “coordinate functions” of each.

\(f: \mathbb{R} \to \mathbb{R}^3\), \(f(t) = (2t^2, t^4, e^t\sin{t})\)
\(g: \mathbb{R}^2 \to \mathbb{R}^2\), \(g(x,y) = (2x^2y, x-y)\)
\(h: \mathbb{R}^3 \to \mathbb{R}^4\), \(h(x,y,z) = (x^2+y^2+z^2, xyz, x+y+z, x-y+z)\)

Coordinate functions

Definition.

Let \(f: \mathbb{R}^n \to \mathbb{R}^m\) be a function with \[ (y_1,y_2,\ldots,y_m) = f(x_1,x_2,\ldots,x_n). \]

For each \(i=1,2,\ldots,m\), the \(i\)-th coordinate function of \(f\) is the function \[ f_i: \mathbb{R}^n \to \mathbb{R}, \quad y_i = f_i(x_1,x_2,\ldots,x_n). \]

Coordinate functions and derivatives, part 1

Suppose that \(f: \mathbb{R}^2 \to \mathbb{R}^3\) is a differentiable function. Then its derivative \(f'(x,y)\) is a \(3 \times 2\) matrix, which we suppose has the form \[ f'(x,y) = \begin{bmatrix} d_{11} & d_{12} \\ d_{21} & d_{22} \\ d_{31} & d_{32} \end{bmatrix}, \] where the \(d_{ij}\) are unknowns that (until now) we had to find using the limit definition of the derivative.
The goal is to relate the \(d_{ij}\) to something easier to compute.
To do this, suppose that we write \(f\) in terms of its coordinate functions: \[ f(x,y) = (f_1(x,y), f_2(x,y), f_3(x,y)). \]
Then, with \(\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}\), we have \[ f'(x,y) \mathbf{e}_1 = \lim_{\lambda \to 0} \frac{f\left( \begin{bmatrix} x + \lambda \\ y \end{bmatrix} \right) - f\left( \begin{bmatrix} x \\ y \end{bmatrix} \right)}{\lambda} =\begin{bmatrix} \lim\limits_{\lambda \to 0} \displaystyle\frac{f_1(x + \lambda, y) - f_1(x,y)}{\lambda} \\ \lim\limits_{\lambda \to 0} \displaystyle\frac{f_2(x + \lambda, y) - f_2(x,y)}{\lambda} \\ \lim\limits_{\lambda \to 0} \displaystyle\frac{f_3(x + \lambda, y) - f_3(x,y)}{\lambda} \end{bmatrix} =\begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x}(x,y) \\ \displaystyle\frac{\partial f_2}{\partial x}(x,y) \\ \displaystyle\frac{\partial f_3}{\partial x}(x,y) \end{bmatrix}. \]
On the other hand, we have \[ f'(x,y)\mathbf{e}_1 = \begin{bmatrix} d_{11} & d_{12} \\ d_{21} & d_{22} \\ d_{31} & d_{32} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} d_{11} \\ d_{21} \\ d_{31} \end{bmatrix}. \]
We conclude that \[ \begin{bmatrix} d_{11} \\ d_{21} \\ d_{31} \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x}(x,y) \\ \displaystyle\frac{\partial f_2}{\partial x}(x,y) \\ \displaystyle\frac{\partial f_3}{\partial x}(x,y) \end{bmatrix}. \]

Coordinate functions and derivatives, part 2

If we repeat the same argument with \(\mathbf{e}_2\), we find that \[ \begin{bmatrix} d_{12} \\ d_{22} \\ d_{32} \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial y}(x,y) \\ \displaystyle\frac{\partial f_2}{\partial y}(x,y) \\ \displaystyle\frac{\partial f_3}{\partial y}(x,y) \end{bmatrix}. \]
Thus, we have shown that \[ f'(x,y) = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x}(x,y) & \displaystyle\frac{\partial f_1}{\partial y}(x,y) \\ \displaystyle\frac{\partial f_2}{\partial x}(x,y) & \displaystyle\frac{\partial f_2}{\partial y}(x,y) \\ \displaystyle\frac{\partial f_3}{\partial x}(x,y) & \displaystyle\frac{\partial f_3}{\partial y}(x,y) \end{bmatrix}. \]
If instead we considered a function \(g: \mathbb{R}^4 \to \mathbb{R}^2\), we would find that \[ g'(x,y,z,w) = \begin{bmatrix} \displaystyle\frac{\partial g_1}{\partial x}(x,y,z,w) & \displaystyle\frac{\partial g_1}{\partial y}(x,y,z,w) & \displaystyle\frac{\partial g_1}{\partial z}(x,y,z,w) & \displaystyle\frac{\partial g_1}{\partial w}(x,y,z,w) \\ \displaystyle\frac{\partial g_2}{\partial x}(x,y,z,w) & \displaystyle\frac{\partial g_2}{\partial y}(x,y,z,w) & \displaystyle\frac{\partial g_2}{\partial z}(x,y,z,w) & \displaystyle\frac{\partial g_2}{\partial w}(x,y,z,w) \end{bmatrix}. \]
Do you see the pattern?

Coordinate functions and derivatives, part 3

Theorem.

Suppose that \(f: \mathbb{R}^n \to \mathbb{R}^m\) is a differentiable function with coordinate functions \(f_1,f_2,\ldots,f_m\). Then, the derivative of \(f\) is given by \[ f'(x_1,x_2,\ldots,x_n) = \begin{bmatrix} \displaystyle\frac{\partial f_1}{\partial x_1}(x_1,x_2,\ldots,x_n) & \displaystyle\frac{\partial f_1}{\partial x_2}(x_1,x_2,\ldots,x_n) & \cdots & \displaystyle\frac{\partial f_1}{\partial x_n}(x_1,x_2,\ldots,x_n) \\ \displaystyle\frac{\partial f_2}{\partial x_1}(x_1,x_2,\ldots,x_n) & \displaystyle\frac{\partial f_2}{\partial x_2}(x_1,x_2,\ldots,x_n) & \cdots & \displaystyle\frac{\partial f_2}{\partial x_n}(x_1,x_2,\ldots,x_n) \\ \vdots & \vdots & \ddots & \vdots \\ \displaystyle\frac{\partial f_m}{\partial x_1}(x_1,x_2,\ldots,x_n) & \displaystyle\frac{\partial f_m}{\partial x_2}(x_1,x_2,\ldots,x_n) & \cdots & \displaystyle\frac{\partial f_m}{\partial x_n}(x_1,x_2,\ldots,x_n) \end{bmatrix}. \]

Exercise 2: Practice computing derivatives of functions

Find the derivative of each of the following functions:

\(f: \mathbb{R} \to \mathbb{R}^3\), \(f(t) = (2t^2, t^4, e^t\sin{t})\)
\(g: \mathbb{R}^2 \to \mathbb{R}^2\), \(g(x,y) = (2x^2y, x-y)\)
\(h: \mathbb{R}^3 \to \mathbb{R}^4\), \(h(x,y,z) = (x^2+y^2+z^2, xyz, x+y+z, x-y+z)\)

Exercise 3: A special function

Consider the function

\[ f: \mathbb{R}^2 \to \mathbb{R}^2, \quad (x,y) =f(r,\theta) = (r\cos{\theta}, r\sin{\theta}). \]

Describe the action of this function on the portion of the \(r\theta\)-plane where \(r>0\) and \(0\leq \theta < 2\pi\).
Find the derivative of this function.