Fdtxjr

In a variety of contexts, I have noticed hints of a strong connection between exact differential equations and machinery from multivariable calculus.

From another question, I have gathered that the geometry of finding the general solution to an exact differential equation $M(x, y) dx + N(x, y) dy = 0$ includes finding the potential function of $beginbmatrixM(x, y) \ N(x, y)endbmatrix$, that is, every surface in $mathbbR^3$ whose gradient is the vector field $beginbmatrixM(x, y) \ N(x, y)endbmatrix$. What is missing from this description? For example, we can write the LHS of a not-necessarily-exact $M(x, y) dx + N(x, y) dy = 0$ as $beginbmatrixM(x, y) \ N(x, y) endbmatrix cdot beginbmatrixdx \ dyendbmatrix$, which expresses the differential of work done by $beginbmatrixM(x, y) \ N(x, y)endbmatrix$, and thus shows promise for extending the above geometric picture. In fact, since work is a path function, I suspect the Wikipedia quote

In mathematics, an integrating factor is a function that is chosen to
facilitate the solving of a given equation involving differentials. It
is commonly used to solve ordinary differential equations, but is also
used within multivariable calculus when multiplying through by an
integrating factor allows an inexact differential to be made into an
exact differential (which can then be integrated to give a scalar
field). This is especially useful in thermodynamics where temperature
becomes the integrating factor that makes entropy an exact
differential.

is misleading, in that the ordinary differential equations and multivariable calculus uses are actually identical, which if correct, might allow geometry from other path functions (such as the referenced thermodynamics example) to be leveraged.

A big part of what I'm trying to remedy is that it's not clear to me why finding the potential function of $beginbmatrixM(x, y) \ N(x, y)endbmatrix$ should solve the ODE. I understand the algebraic rationale that the solution technique undoes the multivariable chain rule (multiply through by $dx$ then integrate both sides), but is the geometric picture just an afterthought, or does it offer its own independent rationale for the solution technique?