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I got the definition from the book :-Partial differential equations,American Mathematical Society,Lawrence C Evans.

Let $U\subseteq \mathbb R^n$ be an open subset. An expression of the form $$F(D^ku(x),D^{k-1}u(x),..., Du(x),u(x),x)=0(x\in U)$$ is called a k-th order partial differential equation, where $$F:\mathbb R^{n^k}\times \mathbb R^{n^{k-1}}\times...\mathbb R^n\times \mathbb R\times U\to \mathbb R$$ is given, and $$u:U\to \mathbb R$$ is the unknown function.

Doubt on the definition. I know, $x\in U\subseteq \mathbb R^n$

$u(x)\in R$

$Du(x)\in R^n$ ($\because$ it is a gradient.)

..........

How does the $D^ku(x)\in \mathbb R^{n^k}$? Could you suggest some readings for more clarity?

Unknown x
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    This is explained in Appendix A, p. 701. Unfortunately it is not quite correct: $D^k u(x)$ is not the set of all partial derivatives of order $k$, but the indexed family $(D^\alpha u(x))_{\lvert \alpha \rvert = k}$. This can be identified with an element of $\mathbb R^{n^k}$ since there are $n^k$ multiindices $\alpha$ of order $k$. – Paul Frost Oct 31 '22 at 11:33
  • Why $|\alpha|?$ – Unknown x Nov 01 '22 at 07:58
  • This is defined on p. 701. We have $\alpha = (\alpha_1,\ldots,\alpha_n)$ where the $\alpha_i$ are integers $\ge 0$ and $\lvert \alpha \rvert = \alpha_1+\ldots +\alpha_n$. The indexed family for order $k$ consists of all $D^\alpha u(x)$ with $\lvert \alpha \rvert = k$. – Paul Frost Nov 01 '22 at 09:25
  • Only 664 pages in this text. – Unknown x Nov 01 '22 at 10:19
  • I used the second edition which has 749 pages. http://home.ustc.edu.cn/~xushijie/pdf/textbooks/pde-evans.pdf – Paul Frost Nov 01 '22 at 10:59

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Briefly: as you know, $Du(x) \in \mathbb{R}^n$. I.e., you can express the differential of a function $u$ as a linear combination of $\frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n}$.

For $D^2u(x)$, you need to consider all the combinations $\frac{\partial^2}{\partial x_i \partial x_j}$. This is $n^2$. For any $k$, how many combinations of $\frac{\partial^k}{\partial x_{i_1}\dots \partial x_{i_k}}$ do you have? This is $n^k$.

I suggest you to double-check multivariate calculus. Take a look to this post, where there are many good references suggested: References for multivariable calculus

Jotabeta
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