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I was going through the introductory chapter on PDE, and the following highlighted text does not make any sense to me:

$$F\left(x,y,z,\frac{\partial z}{\partial x},\frac{\partial z}{\partial y}\right)=0\tag 1$$ A solution of Eq. (1) in some domain $\Omega$ of $\mathbb{R} ^2$ is a function $z = f(x, y)$ defined and is of C' in $\Omega$ should satisfy the following two conditions:

In my current capacity I think that this highlighted text is incomplete, and hence not making sense. If not please tell me what does it mean? If incomplete please let me know what the author was trying to convey there.

The image of the whole page is attached below, for further reference. An except from book Introduction to Partial Differential Equations By K. Sankara Rao

Shyam Tripathi
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  • What you quote is surely incomplete as it ends with "should satisfy the following two conditions:" . What exactly in that screenshot does not make sense to you? – Kurt G. Apr 17 '23 at 13:12
  • The phrase, "defined and is of", what does this mean? Like is it some kind of mathematical connective in the likes of "if and only if", etc? – Shyam Tripathi Apr 17 '23 at 13:21
  • We do not have to ponder about the beauty of that author's English. What counts is that the solution of the PDE is a function $z=f(x,y)$ which is differentiable. That's all. – Kurt G. Apr 17 '23 at 13:25
  • Oh. Thanks a lot. I thought it had some mathematical meaning behind it! – Shyam Tripathi Apr 17 '23 at 13:27

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