: What aspects of this diagram should I change to make it suitable as a modern cover image of a textbook? I want to use the following diagram on the cover image of a textbook: What aspects

@Turnbaugh909

I think that the diagram conveys a solemn style, which most probably fits the the content of your book and would be destroyed by many conceivable “modernisations”. As a normal textbook illustration, it is almost perfect, and you should keep the aspect that make it such on the cover.
The most important quality of a math textbook is the intelligibility of the content and this is something your cover should convey – rather than being “fancy”.
In particular I would advise against:


Using some sort of flat design for the manifold (upper left): This would reduce the intelligibility of this part. While this does matter for the cover per se, you want to avoid the impression that all your illustrations are like this.
Using “more 3D” for the manifold. While this may not impede the intelligibility, when well done, it overemphasises this part, as you cannot apply a similar effect to the other components, without this seeming artificical (what it would be).
Using some sort of gradient, e.g., to make the manifold, ℝm or ℝ seem unbounded. This would not match with the solid black-and-white style you are using elsewhere, which would be difficult to change. Also it probably would destroy the simplicity of the diagram.


That being said, there is one main aspect I would change for the cover and this is the composition of elements. While it is alright within the textbook where the diagram should not occupy too much space (not only for economic reasons but also because you want to avoid the reader having to turn pages more than necessary) and is surrounded by text, on the cover you usually have a roughly quadratic space available, which you should use. In detail:


I would centre ℝ horizontally such that the centres of the three main components describe an equiliteral triangle. Right now, there is too much focus on the left side of the diagram.
I suggest trying out to make the arrows connecting the components such that they all lie on a circle or ellpsis, which is horizontally centred on the diagram. I cannot tell whether this will actually look good, though.
Position the legend for the arrows (x, f, f∘x-1) such that they are not too close to the three main components, e.g., x is rather close to ℝm.
Try positioning ℝm such that the ℝ is centred above the square below. I cannot tell whether this will actually look good, though.
Position the legend for U on the left of U, where there is more free space. Position the legend x(U) on the top right of x(U), and position the legend for ℝ below it centred between the two heads of the arrows pointing to ℝ. This achieves a certain symmetry with the arrows, the objects they are connecting and the legends of these objects.


As for colour, I would suggest only to use two or three colours and only to distinguish certain types of elements. For example you could colour the arrows for the mappings and the corresponding legend differently from the rest and leave it at that. I would not use a different colour for the grids, as it would put too much emphasis on them.

Some further remarks:


The “borders” of M and ℝm are very prominent¹. I would either decrease their thickness or (probably better) increase the thickness of ℝ and the border of U.
The ℝ does not seem to match the other symbols in type colour¹.
The arrowheads are very prominent. I would rather use non-solid arrowheads instead, e.g., like this:






¹ This may be an effect of the low-resolution rendering, though. Confirm this before continuing.

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