Testing exactness · Differential Equations

Intuition

Finding a potential to prove an equation is exact would be going round in a circle. There is a test that needs two derivatives. If M and N are the rates of change of a potential F in x and in y, then the rate of M in y and the rate of N in x are the two mixed derivatives of F, taken in opposite orders, and for a smooth function those agree. So an exact equation must pass that test. On a rectangle, or any region without holes, the converse is true as well, and the test is decisive.

Checking a set of directions for consistency before drawing the map. If going east then north and going north then east from the same corner disagree about how far you have climbed, no single height map can produce those directions.

Two derivatives that must agree

If $M$ , $N$ and their first partial derivatives are continuous on a rectangle, then $M\,dx+N\,dy=0$ is exact there if and only if $M_{y}=N_{x}$ . The forward direction is the symmetry of mixed partial derivatives, $F_{xy}=F_{yx}$ , a theorem of multivariable calculus taken here on trust. The backward direction is proved by building the potential, which is the next lesson.

M_{y}=N_{x}\ \Longleftrightarrow\ \text{exact on a rectangle}

Using the test

$(2xy+3)\,dx+(x^{2}-1)\,dy=0$ : , so it is exact.

Exact implies the test

F_{x}=M,\ F_{y}=N\ \Longrightarrow\ M_{y}=N_{x}

Differentiate the first condition in y and the second in x. The left-hand sides become the two mixed partial derivatives of F, in opposite orders. For a function with continuous second derivatives these are equal, a theorem of multivariable calculus this course uses without proof. So the crossed derivatives of M and N are the same function.

Proof steps

F_{x}=M,\qquad F_{y}=N

Start from a potential.

M_{y}=F_{xy},\qquad N_{x}=F_{yx}

Differentiate the first in y and the second in x.

F_{xy}=F_{yx}

Mixed partial derivatives of a smooth function agree, whichever order they are taken in.

M_{y}=N_{x}

So an exact equation always passes the test.

Applications

Practice

Crossed Derivatives

Differentiate the coefficient of $dx$ in $y$ and the coefficient of $dy$ in $x$ . Equal means exact, on a rectangle.

M_{y}=N_{x}

Try it

Is $(3x^{2}+2y)\,dx+(2x+4y)\,dy=0$ exact?