Need help understanding these how this differential equations was solved

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Hello community members

I have been trying to understand differentials and came across this notes which explains inexact differentials. BUt having trouble grasping it. Could someone help me understand these 2 questions?

1)Here, Equation 145 is given to be a gradient giving a direction in x-y plane.But I think the equation just gives a slope as it does not appear to be a vector.

Equation 145

2)Here in equation 147

Equation 147

how is

$$Y′∂σ/∂x=X′∂σ/∂y=X′Y′/τ$$

ie how did they arrive at $$X′Y′/τ$$

Would be really grateful if someone could help me figure this out, please.

Thank you.