Given a vector field $F:$

$$ \vec{F} = \begin{bmatrix} F_1(x,y,z) \\ F_2(x,y,z) \\ F_3(x,y,z) \\ \end{bmatrix} : \mathbb{R}^3 \to \mathbb{R}^3, $$

each component $F_i$ is itself a scalar field, and so we can take the gradient of each one:

$$ \nabla F_1 = \begin{bmatrix} \dfrac{\partial F_1}{\partial x} \\ \dfrac{\partial F_1}{\partial y} \\ \dfrac{\partial F_1}{\partial z} \\ \end{bmatrix}, \quad \nabla F_2 = \begin{bmatrix} \dfrac{\partial F_2}{\partial x} \\ \dfrac{\partial F_2}{\partial y} \\ \dfrac{\partial F_2}{\partial z} \\ \end{bmatrix}, \quad \nabla F_3 = \begin{bmatrix} \dfrac{\partial F_3}{\partial x} \\ \dfrac{\partial F_3}{\partial y} \\ \dfrac{\partial F_3}{\partial z} \\ \end{bmatrix}. $$

Now, the jacobian of $\vec{F}$ is the @matrix that contains all of these gradients:

$$ \vec{J} = \nabla \vec{F} = \vec{F} = \begin{bmatrix} (\nabla F_1)^\mathsf{T} \\ (\nabla F_2)^\mathsf{T} \\ (\nabla F_3)^\mathsf{T} \\ \end{bmatrix} = \begin{bmatrix} \dfrac{\partial F_1}{\partial x} & \dfrac{\partial F_1}{\partial y} & \dfrac{\partial F_1}{\partial z}\\ \dfrac{\partial F_2}{\partial x} & \dfrac{\partial F_2}{\partial y} & \dfrac{\partial F_2}{\partial z}\\ \dfrac{\partial F_3}{\partial x} & \dfrac{\partial F_3}{\partial y} & \dfrac{\partial F_3}{\partial z}\\ \end{bmatrix}. $$

Here:

The @rows are the @transposes of gradients of each component of $\vec{F}.$
The @columns tell how each @coordinate direction affects all components of $\vec{F}.$

Thus the jacobian tells us how each component of $\vec{F}$ changes with every @coordinate. It can be viewed as the "gradient of a vector field."

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