Discussion
The condition for \( M\,dx + N\,dy = 0 \) to be exact is
Answer: B
Exactness means the expression is \( d\phi \) for some \( \phi \), with \( M = \phi_{x} \) and \( N = \phi_{y} \). Equality of the mixed second derivatives then requires
\[ \frac{\partial M}{\partial y} \;=\; \frac{\partial N}{\partial x}. \]
The cross pairing is what matters: comparing each with the derivative in its own variable tests nothing about a potential
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