Does This Equation Represent a Family of Surfaces in 3-D Space?

[Red_CCF]

Hi

I am currently reading a book where this showed up:

The author gave a $3$ parameter equation (note $Y$ and $y$ are two separate variables):

Y_1dy_1 + Y_2 dy_2 + Y_3dy_3 = 0

and states that this does not necessarily represent a family of surfaces in 3-D space and that only if the coefficient in the above equation satisfies (edit: the equation below should be partial derivatives, I can't have it changed for some reason):

Y_1\left(\frac{dY_2}{dy_3} - \frac{dY_3}{dy_2}\right) + Y_2\left(\frac{dY_3}{dy_1} - \frac{dY_1}{dy_3}\right) + Y_3\left(\frac{dY_1}{dy_2} - \frac{dY_2}{dy_1}\right) = 0

Edit (Mark44): Is this what you meant?
$$Y_1\left(\frac{\partial Y_2}{\partial y_3} - \frac{\partial Y_3}{\partial y_2}\right) + Y_2\left(\frac{\partial Y_3}{\partial y_1} - \frac{\partial Y_1}{\partial y_3}\right) + Y_3\left(\frac{\partial Y_1}{\partial y_2} - \frac{\partial Y_2}{\partial y_1}\right) = 0 $$
I don't know the answer to your question, but thought I would edit your post for you.
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can the integral result in a family of surfaces. The example he gave was:

y_1 dy_1 + y_2dy_2 + y_3dy_3 = 0

for which gives an set of spheres.

I have no idea how he got from the first to the second equation. Can anyone help me out?

Thanks

 

[Red_CCF]

Hi Mark44

Yes the highlighted is what I meant, thank you for changing it