Definition

For an open set U⊆ℝnU \subseteq \mathbb{R}^n a Cᵏ-function is a continuous function f:U→ℝf: U \to \mathbb{R} if all of its partial derivatives of order at most kk exist and are continuous on UU, with

∂αf:=∂|α|f∂xα:=∂|α|f(∂x1)α1...(∂xn)αn\partial^\alpha f := \frac{\partial^{|\alpha|} f}{\partial x^\alpha} := \frac{\partial^{|\alpha|} f}{(\partial x^1)^{\alpha_1}...(\partial x^n)^{\alpha_n}}

where |α|=α1+...+αn≤k| \alpha | = \alpha_1 + ... + \alpha_n \leq k.

References

  1. http://staff.ustc.edu.cn/~wangzuoq/Courses/18F-Manifolds/Notes/Lec02.pdf