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Moving on with Stepanov's book I have reached the subject equations which have the following form (3 variables):

Where P, Q, R are sufficiently differentiable functions of x,y,z.

Excercise 205:

Integrating factor:

Will we always be lucky to have an appropriate factor to cast the equation into a full differential form? The book gives a negative answer setting out a very specific condition on the coefficients.

Assume the equation does have a solution and this solution is a 2-dimensional manifold, i.e has the form:

or (locally, at least):

Then

On the other hand, by virtue of the equation (assuming R does not vanish identically):

Comparing the coefficients:

This is an overdetermined system: one function, two equations which generally does not have a solution The integrability condition can be obtained by equation mixed second derivatives, however I will quote the geometrical argument which may also shed light on some fact presented below.

Consider an infinitesimal shift along the manifold from to . Then z will take value:

From here we move to the point with coordinates without leaving the manifold. New value of z:

Similarly, if we first move along and then along ,  we arrive at the following point:

Now we require that whatever route is chosen it leads to the same point on the manifold (up to the terms of the second order). This leads to the following equation:

Now that was the book, here are some thoughts about this theory.

1) First, equation   definitely points to some categories of vector analysis. Indeed, the factors of P, Q and R are the components of the rotor of the vector field . Hence, the condition can be rewritten in a more compact form:

At first sight this should hold trivially for any , for the rotor is by definition perpendicular to the plane defined by and tangent . However, this would only be true, if the solution were indeed an 2-dimensional manifold. If there is no such solution, then the whole derivation becomes invalid.

2) There is another reason why I prefer the geometric argument over comparing the mixed derivatives. The logic is very similar to that used to derive Cauchy-Riemann conditions for the analytic function. Remarkably enough, we can also apply complex formalism to the above problem. Consider the following operator:

where .

Assuming again that the solution exists in the form and using the above shortcuts for partial derivatives we obtain:

Now apply to both parts, where is complex conjugate:

stands for "full partial derivative" where dependance of on is taken into account. Replacing and with their values, we obtain:

where is the Laplacian, On the other hand:

Hence

which gives the above integrability condition.

So this is another example of how recourse to complex values can reveal deep facts behind the otherwise unfamiliar looking expressions. And formulate them in a nice compact form as well.

In conclusion here is an example where integrability condition does not hold:

To solve it we rewrite it as follows:

Now let

where is an arbitrary function. Then

These 2 relations give the general solution.

Strolling struggling on:

This is an inhomogeneous equation. Following the theory we try to
find the general solution in an implicit form:

It is proven that the solution found in this form is indeed
general, i.e. we are not losing any solutions on the way.

Now we can write the associated system (in the symmetrical form):

Or more conveniently for this example, in the canonical form:

Multiplying these equations by l, m and n respectively and summing we get:

This is one of the first integrals of the system. Now multiplying the equations by x, y and z respectively amd summing we obtain:

Therefore, the general solution has the following form:

Geometrically the first solution represents a plane in a 3d space with angular coefficients of the normal vector . The second integral represents a sphere centered at the origin. Therefore, the characteristics of the equation (the curves,

resulting from intersection of these surfaces) are the circles centered on the line passing through the origin with the above mentioned angular coefficients.

Indeed, another way to look at it is rewrite the equation in the following form:

$$\left|\begin{array}{ccc}

\frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial u}{\partial z}\\

l & m & n\\

x & y & z\end{array}\right|=0$$

Or:

Where is a tangential vector to the surface , is the vector of the axis of revolution and is the radius vector of an arbitrary point on the surface. It means that for every point on the surface the tangent vector must lie in the plane

passing through the axis of revolution. This is natural, for the surface is ontained by rotating a plane curve against the axis.