# 矢量微积分在流体力学中的应用 Vector Calculus With Applications in Fluid Mechanics MATH225

$$\frac{\partial}{\partial x_{k}}\left(\frac{\partial f_{i}}{\partial x_{j}}\right)(\boldsymbol{x})$$
(which we call second-order derivatives of $f_{i}$ ) all exist and are continuous on $U$ for each $i=1, \ldots, m$ and $j, k=1, \ldots, n$. We denote the above second-order derivative by
$$\frac{\partial^{2} f_{i}}{\partial x_{k} \partial x_{j}}(\boldsymbol{x})$$
One can clearly iterate this process in order to define a function of class $C^{p}$ on $U$. If $f$ is of class $C^{p}$ for every $p \in \mathbb{N}$, then we say that it is of class $C^{\infty}$ on $U$.

$$g: U \subset \mathbb{R}^{n} \rightarrow \mathbb{R}$$
be a function of class $C^{1}$ on the open set $U$. Then
$$\mathbf{F}:=\nabla g: U \rightarrow \mathbb{R}^{n}, \mathbf{F}(\boldsymbol{x})=\nabla g(\boldsymbol{x}):=\left(\frac{\partial g}{\partial x_{1}}(\boldsymbol{x}), \ldots, \frac{\partial g}{\partial x_{n}}(\boldsymbol{x})\right)$$
is a vector field and is called the gradient field of $g$.