Jacobian Calculator
Given $m$ differentiable, real-valued functions of $(x_1,x_2,\ldots,x_n)$: \[ \begin{array}{l} f_1(x_1,x_2,\ldots,x_n) \\ f_2(x_1,x_2,\ldots,x_n) \\ \cdots \\ f_m(x_1,x_2,\ldots,x_n) \end{array} \] and given $n$ real numbers $a_1,a_2,\ldots,a_n$, this application calculates at $x_i=a_i$, where $i=1,\ldots,n$, the $m$ function values of $f_1,f_2,\ldots,f_m$ and the $m\times n$ Jacobian matrix: \[ \begin{bmatrix} \frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \dots & \frac{\partial f_1}{\partial x_n} \\ \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \dots & \frac{\partial f_2}{\partial x_n} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial f_m}{\partial x_1} & \frac{\partial f_m}{\partial x_2} & \dots & \frac{\partial f_m}{\partial x_n} \end{bmatrix} \] Note that some literature or wiki Jacobian defines the Jacobian as the transpose of the matrix given above. We use only this form in all calculators that require Jacobian calculation. In the special case $m=1$ the Jacobian is reduced to the gradient of $f$ and, furthermore, if $n=1$ it is just the derivative of $f$ with respect to $x$ or $\frac{df}{dx}$.
Input Data
The calculator required from users two inputs: $f_1(x_1,x_2,\ldots,x_n),\ldots,f_m(x_1,x_2,\ldots,x_n)$ and $a_1,a_2,\ldots,a_n$. Both inputs are the same format as described in Gradient Calculator. The functions $f_1,\ldots,f_m$ must be entered one function per line.