Aren't all equations defining a field?

[LucasGB]

A scalar field is usually defined as a function which assigns a number to every point in space. But aren't all equations assigning a number to every point in space? For example, take z = x^2. If I plot it in a 2D graph, I get a line. But if I plot it in a 3D graph, I get a surface. So there I have it, I can safely say this equations defines a scalar field in 2D space.

The point I'm trying to make is this. Every function of one and two variables can be treated as a function of three variables, where the third variable doesn't have any importance. Every curve can be extended into the third dimension to form a surface. So every function is assigning a number to every point in space, and this is a scalar field. Am I right?

 

[Kyouran]

In mathematics, a scalar field isn't restricted to 3D.

From wikipedia:
Mathematically, a scalar field on a region U is a real or complex-valued function on U.

So, in R, a scalar field is any function f:R^n -> R. n does not necessarily have to be 3.

In physics however, when talking about a scalar field, people usually mean a function f:R^3 -> R. An example is the temperature in a room.

 

[LucasGB]

Yes, you're right. I guess my question ultimately comes down to this: is it possible to define a scalar field in two dimensions through a function of one variable? I believe it is.

For example, z = x + y is a function of two variables and certainly defines a scalar field in two dimensions. But z = x + 0y is a function of one variable and it also defines a scalar field in two dimensions. Therefore, two dimensional scalar fields can be defined by functions of one variable, and so can three dimensional scalar fields, and four dimensional scalar fields, etc. because one can always define the field through an equation such as s = x + 0y + 0z +..., which are, nevertheless, functions of one variable.

 

[Mark44]

A scalar field is usually defined as a function which assigns a number to every point in space. But aren't all equations assigning a number to every point in space? For example, take z = x^2. If I plot it in a 2D graph, I get a line. But if I plot it in a 3D graph, I get a surface. So there I have it, I can safely say this equations defines a scalar field in 2D space.

The point I'm trying to make is this. Every function of one and two variables can be treated as a function of three variables, where the third variable doesn't have any importance. Every curve can be extended into the third dimension to form a surface. So every function is assigning a number to every point in space, and this is a scalar field. Am I right?

Maybe I'm missing something here, but if you plot the equation z = x² in 2D (assuming x and z axes), you don't get a line -- you get a parabola. And the parabola isn't defined at every point the the 2D plane, but only at points (x, z) for which z = x². In particular, no points of the graph lie below the x-axis.

If you plot the same equation in 3D space, you do get a surface. More precisely, you get a "cylinder" that is parabolic in cross section, sort of a trough shape. This surface is not defined at every point in 3D space, just at those points (x, y, z) for which z = x². No part of the surface lies below the x-y plane.

 

[LucasGB]

Maybe I'm missing something here, but if you plot the equation z = x² in 2D (assuming x and z axes), you don't get a line -- you get a parabola. And the parabola isn't defined at every point the the 2D plane, but only at points (x, z) for which z = x². In particular, no points of the graph lie below the x-axis.

If you plot the same equation in 3D space, you do get a surface. More precisely, you get a "cylinder" that is parabolic in cross section, sort of a trough shape. This surface is not defined at every point in 3D space, just at those points (x, y, z) for which z = x². No part of the surface lies below the x-y plane.

I expressed myself very poorly. I think I have made my question a little bit clearer in my last post, please take a look at that. Thank you all for your patience.

 

[Kyouran]

Yes, you're right. I guess my question ultimately comes down to this: is it possible to define a scalar field in two dimensions through a function of one variable? I believe it is.

For example, z = x + y is a function of two variables and certainly defines a scalar field in two dimensions. But z = x + 0y is a function of one variable and it also defines a scalar field in two dimensions. Therefore, two dimensional scalar fields can be defined by functions of one variable, and so can three dimensional scalar fields, and four dimensional scalar fields, etc. because one can always define the field through an equation such as s = x + 0y + 0z +..., which are, nevertheless, functions of one variable.

Ofcourse you can. Sometimes people write f(x) = 3. This is a function of x. But the value of the function does not depend on x at all. Likewise, you can write f(x,y) = 2x, and so on.

If I have a set X={0,1,2,...} and a set with only an element '3' in it, you can write a function f(x) = 3 that shows the relationship between the elements of these sets, even though there doesn't appear to be any sort of real relation between them.

 

[LucasGB]

Ofcourse you can. Sometimes people write f(x) = 3. This is a function of x. But the value of the function does not depend on x at all. Likewise, you can write f(x,y) = 2x, and so on.

If I have a set X={0,1,2,...} and a set with only an element '3' in it, you can write a function f(x) = 3 that shows the relationship between the elements of these sets, even though there doesn't appear to be any sort of real relation between them.

OK, you clarified some things for me, but what about this: f(x) = 3 can be rewritten as f(x) = 3 + 0x, or f(x,y) = 3 + 0x + 0y, ..., and the function would be the same. So I can't say if it's a function of one variable, two variables, three variables, etc. How can you say f(x) = 3 + 0x is a function of one variable (which is x), when f(x,y) = 3 + 0x + 0y is exactly the same function but seems to be a function of two variables (x and y)?

 

[Kyouran]

OK, you clarified some things for me, but what about this: f(x) = 3 can be rewritten as f(x) = 3 + 0x, or f(x,y) = 3 + 0x + 0y, ..., and the function would be the same. So I can't say if it's a function of one variable, two variables, three variables, etc. How can you say f(x) = 3 + 0x is a function of one variable (which is x), when f(x,y) = 3 + 0x + 0y is exactly the same function but seems to be a function of two variables (x and y)?

That's why a function is determined by a domain, a codomain and a graph. The domain of a function determines how many variables you have. A function expresses a relation between sets. Which set you choose for your domain is up to you. You can choose the set of all x, or you can choose the set of all ordered pairs (x,y), and so on. The choice is up to you.

 

[LucasGB]

That's why a function is determined by a domain, a codomain and a graph. The domain of a function determines how many variables you have. A function expresses a relation between sets. Which set you choose for your domain is up to you. You can choose the set of all x, or you can choose the set of all ordered pairs (x,y), and so on. The choice is up to you.

Hmmmm...

So a function is not uniquely defined by its equation? It must also come with "instructions" (the definition of the domain)? I think that pretty much clears up the confusion!

So I can say the function z = x + 0y is a function of two variables, if I specify that the domain is the set of ordered pairs (x,y)? On the other hand, if I specify the domain is the set of all x, then I can say it is a function of one variable? If so, I think I understand everything now.

 

[Kyouran]

Hmmmm...

So a function is not uniquely defined by its equation? It must also come with "instructions" (the definition of the domain)? I think that pretty much clears up the confusion!

So I can say the function z = x + 0y is a function of two variables, if I specify that the domain is the set of ordered pairs (x,y)? On the other hand, if I specify the domain is the set of all x, then I can say it is a function of one variable? If so, I think I understand everything now.

That's correct. A function is usually defined by 3 sets: the domain, the codomain and it's graph. An example:

the function (R,R,{(x,z}|z=x^2}) is a function of 1 variable, while the function (R²,R,{(x,y,z)|z=x^2}) is a function of 2 variables.

Maybe I'm missing something here, but if you plot the equation z = x² in 2D (assuming x and z axes), you don't get a line -- you get a parabola. And the parabola isn't defined at every point the the 2D plane, but only at points (x, z) for which z = x². In particular, no points of the graph lie below the x-axis.

If you plot the same equation in 3D space, you do get a surface. More precisely, you get a "cylinder" that is parabolic in cross section, sort of a trough shape. This surface is not defined at every point in 3D space, just at those points (x, y, z) for which z = x². No part of the surface lies below the x-y plane.

One more thing about the plots, just to make sure there is no confusion with scalar field and such.
If you have a function z = f(x) = x^2 and you plot it in the xz plane, it's true that the parabola is not defined at every point of the 2D plane. But, this does not imply it's not a scalar field anymore, in fact it remains a 1 dimensional scalar field (the x axis), with the z values being the "values at every point"

If we include a 2nd variable y, we still have a scalar field: z = f(x,y) = x^2. Now we have a 2 dimensional scalar field in the xy plane.

 

[LucasGB]

That's correct. A function is usually defined by 3 sets: the domain, the codomain and it's graph. An example:

the function (R,R,{(x,z}|z=x^2}) is a function of 1 variable, while the function (R²,R,{(x,y,z)|z=x^2}) is a function of 2 variables.

Perfect. Now it all fits together. One more thing, though. By its "graph" is the same as by its equation, right?

One more thing about the plots, just to make sure there is no confusion with scalar field and such.
If you have a function z = f(x) = x^2 and you plot it in the xz plane, it's true that the parabola is not defined at every point of the 2D plane. But, this does not imply it's not a scalar field anymore, in fact it remains a 1 dimensional scalar field (the x axis), with the z values being the "values at every point"

If we include a 2nd variable y, we still have a scalar field: z = f(x,y) = x^2. Now we have a 2 dimensional scalar field in the xy plane.

That is what I originally meant. Thank you for clarifying.

 

[Kyouran]

Perfect. Now it all fits together. One more thing, though. By its "graph" is the same as by its equation, right?

Actually it's a set. I don't have much time anymore right now, so forgive me for just posting a link or 2:

http://en.wikipedia.org/wiki/Graph_of_a_function
http://en.wikipedia.org/wiki/Function_(mathematics )