技术问询：若Jacobian为一阶导数、Hessian为二阶导数，三阶导数称什么？

Hey there! Great question—this gets into the less standardized side of multivariable calculus derivatives, which is a fun deep dive. Let's break it down clearly:

First, let's recap the basics to set context:

• The Jacobian is the standard name for the first-order derivative of a vector-valued function (mapping from ( \mathbb{R}^n ) to ( \mathbb{R}^m )). It takes the form of a matrix where each entry is a first partial derivative.
• The Hessian is the standard name for the second-order derivative of a scalar-valued function (mapping from ( \mathbb{R}^n ) to ( \mathbb{R} )). It's an ( n \times n ) matrix of second partial derivatives.

Now, for the third derivative:

• Unlike the Jacobian and Hessian, there isn't a single universally accepted "brand name" that's as widely used. That said, the most common way to refer to it is as the third-order derivative tensor.
• Depending on context, you might also see it described as the Jacobian of the Hessian: since the Hessian (a matrix) can be treated as a vector-valued function when flattened, its Jacobian would correspond to this third-order tensor structure.
• For vector-valued functions, the third derivative becomes a fourth-order tensor, and it's still generally referred to using the same "third-order derivative tensor" label (sometimes with extra context to clarify the function type).

A quick note: The looser naming convention here makes sense because higher-order derivatives beyond the second are far less frequently used in most practical fields like optimization, machine learning, or differential equations—so there hasn't been as much need for a universally agreed-upon shorthand like Jacobian or Hessian.

内容的提问来源于stack exchange，提问作者echo