Level curves, level surfaces, level sets

[Calpalned]

Homework Statement

I know that the equation $z = f(x,y)$ gives a surface while $w = f(x, y, z)$ gives an object that has the same surface shape on top as $z = f(x,y)$ but also includes everything below it. If these statements are correct, what is the level surface of a function of three variables like $F(x,y,z) = k$

Homework Equations

Tangent line for $z = f(z, y)$ = $z-z_0 = f_x$

The Attempt at a Solution

To me, it seems like the level surface of a function of three variables is only a number line. Does this also apply to the level surface of a function of two variables? What about of one variable? Is a level surface the higher dimensional analogy of a level curve, which in itself is a graph of a level set? Finally how are topographical maps related?

Thank you all so much. All of the "level" stuff in calculus is so confusing. What's worse is that because of them, there isn't one single formula for finding a tangent plane...

 

[Ray Vickson]

Homework Statement

I know that the equation $z = f(x,y)$ gives a surface while $w = f(x, y, z)$ gives an object that has the same surface shape on top as $z = f(x,y)$ but also includes everything below it. If these statements are correct, what is the level surface of a function of three variables like $F(x,y,z) = k$

Homework Equations

Tangent line for $z = f(z, y)$ = $z-z_0 = f_x$

The Attempt at a Solution

To me, it seems like the level surface of a function of three variables is only a number line. Does this also apply to the level surface of a function of two variables? What about of one variable? Is a level surface the higher dimensional analogy of a level curve, which in itself is a graph of a level set? Finally how are topographical maps related?

Thank you all so much. All of the "level" stuff in calculus is so confusing. What's worse is that because of them, there isn't one single formula for finding a tangent plane...

As far as I can tell your statement that "$w = f(x, y, z)$ gives an object that has the same surface shape on top as $z = f(x,y)$ but also includes everything below it" is wrong. Where did you ever see such a statement?

BTW: Never, never use the same name f for two different functions $f(x,y)$ and $f(x,y,z)$ in the same sentence or paragraph.

 

[Calpalned]

As far as I can tell your statement that "$w = f(x, y, z)$ gives an object that has the same surface shape on top as $z = f(x,y)$ but also includes everything below it" is wrong. Where did you ever see such a statement?

BTW: Never, never use the same name f for two different functions $f(x,y)$ and $f(x,y,z)$ in the same sentence or paragraph.

I'm confused if surfaces are given by $f(x,y,z)$ or just $f(x,y)$

 

[Ray Vickson]

I'm confused if surfaces are given by $f(x,y,z)$ or just $f(x,y)$

Both are possible. For example, we can have the surface $F(x,y,z) = 0$, where $F(x,y,z) = z - f(x,y)$, or we can just write it as $z = f(x,y)$. We can write $z = \sqrt{25 - x^2 - y^2}$ or $x^2 + y^2 + z^2 = 25$. Sometimes you will be given one form and be asked to figure out the other form. Sometimes a tangent line/plane question will give you one form, but sometimes will give you the other. You need to be able to deal with both forms, and you should never use the formulas for one form when dealing with the other form.

You seem to be really confused about this general subject area, so I wonder what you are using as a textbook or course notes, if anything. I think you are trying to go to far too fast, and are attempting to jump to more advanced topics before mastering the previous ones---a sure recipe for disaster.

There are lots of resources available on-line, and the very best policy is to look at several sources, so if one is confusing to you, you can always look at another to (hopefully) clear up some of the arising issues. However, I am old enough (and old-fashioned enough) to believe strongly that there is no substitute for an actual, physical book, printed on real paper (despite the fact that I mostly read pdf files on the computer nowadays). I am a firm believer that students should still go to the library.

 

[SammyS]

I'm confused if surfaces are given by $f(x,y,z)$ or just $f(x,y)$

BTW: Never, never use the same name f for two different functions $f(x,y)$ and $f(x,y,z)$ in the same sentence or paragraph.

I'll repeat what Ray said:
"Never, never use the same name f for two different functions $f(x,y)$ and $f(x,y,z)$ in the same sentence or paragraph."

 

[HallsofIvy]

Homework Statement

I know that the equation $z = f(x,y)$ gives a surface

Yes, every point in three dimensional space can be written as (x, y, z). Requiring that z= f(x, y) restricts to (x, y, f(x, y)) which depends on the two variables, x and y and so is two dimensional- a surface.

while $w = f(x, y, z)$ gives an object that has the same surface shape on top as $z = f(x,y)$ but also includes everything below it.

I don't know what you are trying to say here. With four variables, we would have to be in four dimensional space, with points written as (x, y, z, w). (x, y, z, f(x,y,z) would be a three dimensional hyper-surface in four dimensions.

If these statements are correct, what is the level surface of a function of three variables like $F(x,y,z) = k$

Yes, with "k" a constant, rather than a variable, we could, theoretically, solve for z to get z= f(x,y) with constant k in "f". That is a surface in three dimensions.

2. Homework Equations

Tangent line for $z = f(z, y)$ = $z-z_0 = f_x$

Was this a typo? Did you mean z= f(x,y)? If so this is a surface and so at each point there is a tangent plane containing an infinite number of tangent lines to the surface. The tangent plane would be given by $z- z_0= f_x(x- x_0)+ f_y(y- y_0)$.

3. The Attempt at a Solution

To me, it seems like the level surface of a function of three variables is only a number line.

Why do you say that? A surface is defined by an equation involving three variables, so that, again at least theoretically, we could solve for one variable in terms of the other two, reducing from (x, y, z) to two variables, a surface. Not one variable so not a line.

Does this also apply to the level surface of a function of two variables?

Not quite, if z= f(x,y), since z is determined by the two variables, x and y, geometrically, it describes a two dimensional object, a surface. Setting z equal to a given constant, f(x,y)= k we could solve for one of the variables in terms of the other so have one variable. An equation in one variable describes a level curve, not a level surface.

What about of one variable? Is a level surface the higher dimensional analogy of a level curve, which in itself is a graph of a level set?

If you have a function of one variable, y= f(x), its graph is, of course, a one dimensional curve in two dimensions. Setting y= to a constant, so f(x)= k gives a function that can be solved for x giving x equal to one or more specific numbers. For example, if $y= x^2$, its graph is a parabola on a two dimensional graph. If y= 4, then x= -1 and 1, the "level set" consisting of two numbers.

Finally how are topographical maps related?

A "topographical map" has curves, circling a mountain top or hill, say, connecting points of equal altitude. If we think of the surface of the mountain as given by z= f(x, y), with z the altitude of point (x,y) on some grid, the "contour lines" being level curves

Thank you all so much. All of the "level" stuff in calculus is so confusing.

You seem to be confused as to "dimensions". "z= f(x,y) or, equivalently, g(x, y, z)= constant defines a two dimensional surface in three dimensions. It has level curves and tangent planes. w= f(x, y, z) or g(x, y, z, w)= constant defines a three dimensional surface in a four dimensional space. It would have two dimensional "level surfaces" and three dimension tangent hyper-planes.

What's worse is that because of them, there isn't one single formula for finding a tangent plane...[/QUOTE]
I don't know what you mean by this. "Level curves" have nothing to do with "tangent planes". And there is "one single formula for finding a tangent plane". The tangent plane to the surface z= f(x,y), at $(x_0, y_0, f(x_0, y_0)$ is given by $z= f_x(x_0, y_0)(x- x_0)+ f_y(y- y_0)$. Of course there are other ways of representing surfaces so other tools for finding tangent planes. Complaining about "there isn't one single formula" is like a carpenter complaining about having too many tools- you can't build anything with just a hammer.

 

[Calpalned]

"z= f(x,y) or, equivalently, g(x, y, z) = constant

How does it work? y= f(x) is equivalent to g(x, y) = constant?

 

[SammyS]

How does it work? y= f(x) is equivalent to g(x, y) = constant?

Let f(x) = 2x + 7 .

Then y = f(x) gives you y = 2x + 7 .

This is the same result you get from g(x,y) = -7, for g(x,y) = 2x - y .