Calpalned said:
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.
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.