Name of vector which includes value and its derivatives What would I call a vector that describes a quantity (Temperature in this case) and its first and second derivatives at a specific location? It seems that a vector like this would have a formal name.
For context, I am solving a BVP for temperature distribution through a cylinder and want to describe the value of the Temperature, its derivative, and second derivative in one vector.
 A: If you have $f:\mathbb R^3\rightarrow \mathbb R$, then the first derivative is called gradient $\nabla f$ and the second derivative is called the Hessian $\nabla^2 f$. The gradient is the direction of quickest increase in temperature (and the negative of the gradient is the direction of quickest decrease). The Hessian is symmetric. The gradient is 3x1 and the hessian is 3x3. I do not believe there is a term that includes all three in one vector, and the dimensions are different so it would be hard to put in one vector. For exapmle, the temperature is just a number. So you probably want something like a list. The first element of the list will be a number, the second element of the list will be a vector, and the third element will be a matrix. There is no name for this as far as I know.
If $f(x,y,z):R^3\rightarrow R$, then $\nabla f=\begin{pmatrix}\frac{\partial f}{\partial x}\\\frac{\partial f}{\partial y}\\\frac{\partial f}{\partial z}\end{pmatrix}$ and
$$h(f(x,y,z))=\begin{pmatrix}\frac{\partial^2 f}{\partial x^2}&\frac{\partial^2 f}{\partial x\partial y}&\frac{\partial^2 f}{\partial x\partial z}\\\frac{\partial^2 f}{\partial y\partial x}&\frac{\partial^2 f}{\partial y^2}&\frac{\partial^2 f}{\partial y\partial z}\\\frac{\partial^2 f}{\partial z\partial x}&\frac{\partial^2 f}{\partial z\partial y}&\frac{\partial^2 f}{\partial z^2}\end{pmatrix}$$
