Edited by Valentin Fadeev, Wednesday, 15 Sept 2010, 23:37
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:
Pfaff equation
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.