I'm puzzled by your definition of the implicit function. Is z = 0, period? Or is there a typo in there? And are you trying to think of z = z(x,y)?
if we have an implicit function eqn1
then we can find the standard derivative eqn 2 and
the partial derivative eqn3
However, I am trying to grasp the idea of having a standard/total derivation and partial derivation of functions as above yet we dont have standard or partial integration (ie the reverse). How is that?
If we integrate back eqn2 wrt to x we get eqn1 with a constant. This constant can be determined if BC's or IC's are known (ie 6) etc...but
if we integrate eqn3 wrt x we can get eqn1 with a constant also but we lose the term. How does one get this term back?
I dont think I understand this fully. Can anyone shed some light?
Actually, in that case, your "constant" of integration with respect x is an unknown function of y, which you can then determine sometimes by integrating w.r.t. y and comparing to the original function. Everything here depends on knowing what is a function of what. In the first case, you're treating the function z like this: z(x,y(x)). In the second case, you have z(x,y), and y is not treated like a function of x.if we integrate eqn3 wrt x we can get eqn1 with a constant also but we lose the term. How does one get this term back?
So in reality, there are two different kinds of integration, corresponding to your total and partial derivatives. The difference will play out in the "constants" of integration. Make sense?
So in the first case the y(x) you are referring to is the term?
If I understand correctly then, the method used to determine the constants will reflect the nature of integration ie partial or total?
I have never come across a problem like the first case to determine the constants. Can an example of this be provided?
Thanks for your explanation. Makes it a lot clearer.
I'm saying that if I were given
and told to integrate with respect to x, I would produce the following:
If I were given and told to integrate with respect to x, I would produce the following:
where is an unknown function of y. To find that function, I would need more information. If given the correct information, I could recover the of the original equation.
You see how this works? In the second case, I have to include an unknown function of y, because in taking the partial derivative of the resulting integral with respect to x, I would eliminate that whole term.
Integrating dz/dx, I get for each term
the dx's cancel
How does the third term go to 0 because I get
the dx's cancel
All of these term plus a constant etc. However, I now have two '5xy' terms. Should only have one. What have i done wrong?
I feel stupid now