Jacobian
Basic Definition
A Jacobian is similar to a gradient but for a vector of functions (it is therefore a vector of gradients).
Example:
\begin{align*}
f(x,y) = 3x^2y \\
g(x,y) = 2x+y^8 \\
\mathbf{J} = \bigg[\begin{matrix}\nabla f(x,y) \\ \nabla g(x,y)\end{matrix}\bigg] = \Bigg[\begin{matrix}\frac{\partial f(x,y)}{\partial x} & \frac{\partial f(x,y)}{\partial y}\\ \frac{\partial g(x,y)}{\partial x} & \frac{\partial g(x,y)}{\partial y}\end{matrix}\Bigg]= \bigg[\begin{matrix}6yx & 3x^2 \\ 2 & 8y^7\end{matrix}\bigg]
\end{align*}
NOTE Notation used is the "numerator layout" although many use denominator layout which is tranposed.
\[\bigg[\begin{matrix}6yx & 2 \\ 3x^2 & 8y^7\end{matrix}\bigg] \text{ vs } \bigg[\begin{matrix}6yx & 3x^2 \\ 2 & 8y^7\end{matrix}\bigg]\]