Why are implicit functions used instead of explicit?

[Tosh5457]

Hi, this is a very basic question. I want to know why are implicit functions used, when the purpose of functions is to calculate the independent variable.

For example, why would someone write ay + bx + c = 0 as a "straight line function" (sorry for the possible abuse of language, I don't even know if this is how it's said in english), instead of y = ax + b?

 

[Studiot]

There are two main reasons why you might write down an implicit function.

Firstly you might be forced to do so because you cannot separate the variables

y⁵+2x⁴+3x³y²+15 = 0

Secondly you might be going to solve a system of linear equations by matrix methods so you assemble all the variables on one side of the equation and all the constants on the other.

go well

 

[Hurkyl]

I want to know why are implicit functions used

Implicit relationships are often easier to use than other forms. And sometimes they cannot be rewritten as one variable being a function of the other.

 

[Tosh5457]

Ok good enough That's also the reason there are so many differential/integral equations on physics, because it's not possible to express relationship between variables without these, right?

 

[TylerH]

Ok good enough That's also the reason there are so many differential/integral equations on physics, because it's not possible to express relationship between variables without these, right?

I may be wrong, but I think integrals and differentials are common in physics because motion is common in physics. Integrals and differentials allow one to relate different types of information about motion in ways algebra can't. Like gravitation, for example, with integration, you can express the attraction between objects in terms of their distance. (http://en.wikipedia.org/wiki/N-body_problem#Mathematical_formulation_of_the_n-body_problem)

 

[hunt_mat]

Because there may be no 1-1 relationships, take the definition of the unit circle, x^2+y^2=1, this has all the information that you require but y=root(1-x^2) does not.

 

[HallsofIvy]

Hi, this is a very basic question. I want to know why are implicit functions used, when the purpose of functions is to calculate the independent variable.

This is an unwarrented assumption. Functions are used for many purposes.

For example, why would someone write ay + bx + c = 0 as a "straight line function" (sorry for the possible abuse of language, I don't even know if this is how it's said in english), instead of y = ax + b?

ay+ bx+ c= 0 makes it especially easy to find the two intercepts, (-c/b, 0) and (0, -c/a).

Also there are non-function relations, such as x^2+ y^2= r^2, where we cannot write y as a function of x.

 

[arildno]

Furthermore, the equation y=ax+b is NOT general enough to describe ALL straight lines in the (x,y)-plane.

Vertical lines in the plane (x=constant) cannot be described as a special case of y=ax+b, whereas Ax+By=C IS sufficient for that purpose (B=0)

In general, implicit relationships between variables hold the options open for which variable, in a particular region, CAN be regarded as a function of the other.

 

[Tosh5457]

Thanks for the replies

But many times it's desirable to express a function explicitly. And usually aren't there operators that do the inverse operation of another operator?

For example on Ampère-Maxwell's integral form equation:

Isn't it possible to take out the closed line integral from the left side of the equation, and then the internal product B . ds to get the equation in the form B = ... ?

 

[Hurkyl]

Thanks for the replies

But many times it's desirable to express a function explicitly. And usually aren't there operators that do the inverse operation of another operator?

"Most" operators aren't invertible. Often the solution method involves finding a particular solution and a way to compute other solutions knowing it.

For example, you "invert" the equation

curl B = x​

by first finding one particular solution B₀. Then, all you know is that

B = B₀ + V​

where V is some unknown irrotational vector field.

 

[Tosh5457]

"Most" operators aren't invertible. Often the solution method involves finding a particular solution and a way to compute other solutions knowing it.

For example, you "invert" the equation

curl B = x​

by first finding one particular solution B₀. Then, all you know is that

B = B₀ + V​

where V is some unknown irrotational vector field.

Ok, thanks for the explanation :)