Why is velocity gradient not called a velocity Jacobian? I started thinking about the rate of deformation of fluid in the boundary layer. but here we consider only one of the components of the velocity vector (which is a scalar). But what about just general velocity field and what would its gradient be. Googling it gave me answers that looked like Jacobians and hence my question. Why is the velocity gradient not called a velocity jacobian? Won't the gradient of the velocity vector be a Jacobian? 
 A: The velocity vector gradient is a rank-2 tensor of the form
$$\nabla\mathbf{v}=\begin{bmatrix}\frac{\partial v_x}{\partial x}&\frac{\partial v_x}{\partial y}&\frac{\partial v_x}{\partial z}\\ \frac{\partial v_y}{\partial x}&\frac{\partial v_y}{\partial y}&\frac{\partial v_y}{\partial z}\\ \frac{\partial v_z}{\partial x}&\frac{\partial v_z}{\partial y}&\frac{\partial v_z}{\partial z}\end{bmatrix}$$
and as such, it has the same form as the Jacobian, in that it's a matrix of partial derivatives of a vector-valued function. In fact, the Jacobian is even sometimes called the generalization of the gradient. But your question asks why we call such a quantity the velocity gradient instead of the velocity Jacobian.
Though I'm not sure that there is a concrete answer to this, one possible rationale is that the Jacobian, in physics, is usually used in the context of coordinate transformations, whereas the velocity gradient is usually used to describe the dynamics of a vector field. The two terms carry different physical meanings, even though they are the same mathematical object.
For an analogous situation elsewhere in physics, local hydrostatic pressure and local temperature are both the same mathematical object: they're both scalar fields. If I were to give you a random scalar field without telling you anything about it, you wouldn't be able to tell me if it described the local temperature or the local hydrostatic pressure. But obviously those two scalar fields carry very different physical meaning, and so we distinguish them.
