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The point is to illustrate the univariate functions we get when we take a slice of a function in a given direction, and we're not restricted to the $x$ and $y$ directions.
(PS: I hope that my constant introduction of Mathematica code will encourage everyone to get a copy for themselves! Although yesterday's computer malfunction may tell you that you're better off like Archimedes, just writing in the sand....)
We shouldn't be a stick in the mud about the orientation of our axes: why $x$ and $y$, and not some other pair of directions which are mutually perpendicular? Perhaps we are interested in the slope of the surface along some direction other than $x$ or $y$: hence the idea behind directional derivatives. At a given point at which a function is differentiable, one natural choice for two directions might be the direction in which the function is increasing fastest, and the direction perpendicular to this.
But any direction will do -- take a look at p. 957, Figure 3.
This is partially what motivated me to create that "slicing" demo in Mathematica.
One of the biggest pieces of news is that we're going to be working with vectors -- e.g. $u$ -- and with vector-valued functions; and that will place another demand upon your visualization skills.
But gradients are just another kind of multivariate function: one where the domain is points in the plane, like many of our other functions, but the range is the set of vectors.
At each point in the plane there is a gradient vector pointing -- indicating what?
Are you comfortable with