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$$h(x,y)=\frac{100}{(1+(x+y^2))^2}.$$

The river also passes through $(0.5, 0.01)$.

If one then works out the gradient for this you get $g(x,y)(i+2yj)$, where I've used $i$ and $j$ as the unit vectors.

From this the solutions go on to say that the $g(x,y)$ factor is extraneous and so they ignore it and then conclude that the equation of the river is the solution to the DE $$\frac{dy}{dx}=2y.$$ What I don't understand is how they can just ignore the whole $g(x,y)$ factor. Sorry for the messiness in this, I'm not yet sure how to use latex on this site.

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The gradient is $$\frac{g(x,y)2y}{g(x,y)}=2y,$$

We can see that it appears in both the numerator and denominator and it gets cancelled out.

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