Define the Jacobian matrix of a mapping \(f: \mathbb{R}^n \to \mathbb{R}^m\) at a point \(a = (a_1, \dots, a_n)\) of \(\mathbb{R}^n\). Compute the Jacobian matrix of \(f(r,\theta) = (r\cos\theta, r\sin\theta)\) at \(a = (0,0)\).
Solution
1. Definition of the Jacobian matrix:
1.1. Let \(f: \mathbb{R}^n \to \mathbb{R}^m\) be given by coordinate functions \(f(x) = (f_1(x), f_2(x), \dots, f_m(x))^T\) for \(x = (x_1, \dots, x_n) \in \mathbb{R}^n\).
vector-valued function definition.
1.2. If the partial derivatives \(\frac{\partial f_i}{\partial x_j}(a)\) exist at \(a \in \mathbb{R}^n\) for all \(1 \le i \le m\) and \(1 \le j \le n\), the Jacobian matrix \(J_f(a) \in M_{m \times n}(\mathbb{R})\) is defined as:\[ J_f(a) = \begin{pmatrix} \frac{\partial f_1}{\partial x_1}(a) & \frac{\partial f_1}{\partial x_2}(a) & \cdots & \frac{\partial f_1}{\partial x_n}(a) \\ \frac{\partial f_2}{\partial x_1}(a) & \frac{\partial f_2}{\partial x_2}(a) & \cdots & \frac{\partial f_2}{\partial x_n}(a) \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1}(a) & \frac{\partial f_m}{\partial x_2}(a) & \cdots & \frac{\partial f_m}{\partial x_n}(a) \end{pmatrix}. \]
definition of total derivative matrix representation with respect to standard bases.
2. Compute the Jacobian matrix for \(f(r, \theta) = (r\cos\theta, r\sin\theta)\):
2.1. The component functions are \(f_1(r, \theta) = r\cos\theta\) and \(f_2(r, \theta) = r\sin\theta\).
given mapping.
2.2. Compute the partial derivatives with respect to \(r\) and \(\theta\):\[ \frac{\partial f_1}{\partial r} = \cos\theta, \qquad \frac{\partial f_1}{\partial \theta} = -r\sin\theta, \]\[ \frac{\partial f_2}{\partial r} = \sin\theta, \qquad \frac{\partial f_2}{\partial \theta} = r\cos\theta. \]
product and trigonometric differentiation rules.
2.3. The general Jacobian matrix at \((r, \theta)\) is:\[ J_f(r, \theta) = \begin{pmatrix} \cos\theta & -r\sin\theta \\ \sin\theta & r\cos\theta \end{pmatrix}. \]
assembling partial derivatives into \(2 \times 2\) matrix.
3. Evaluation at \(a = (0, 0)\):
3.1. Setting \((r, \theta) = (0, 0)\):- \(\cos(0) = 1\),- \(-0 \cdot \sin(0) = 0\),- \(\sin(0) = 0\),- \(0 \cdot \cos(0) = 0\).
trigonometric values \(\cos(0) = 1, \sin(0) = 0\).
3.2. Thus:\[ J_f(0, 0) = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}. \]
substituting values from 1 into 3.
4. Q.E.D.
Step 1 gives the definition, and steps 2 and 3 give \(J_f(0,0) = \boxed{\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}}\).