Multivariable function notations

In summary, we understand that there may be some confusion with the notation of functions in multiple variables. It is important to clarify the meaning of "graph" and the convention of writing z=f(x,y). Additionally, when taking the gradient of a function in three-dimensional space, it is important to consider the third variable z as independent unless specified otherwise.
  • #1
medwatt
123
0
Hello,
I've been having some trouble getting some notations straight and hence my question.
Usually when I see f(x,y) it means to me there is some variable z produced for any combination of x and y in the domain of the function. So, z=x^2+y^2 I imagine as a paraboloid.
So z=f(x,y) ... is a function of three variables with x and y being independent . . . but sometimes I see u=F(x,y,z)=f(x,y)-z . . . all of a sudden it seems as though z has suddenly become independent and there are now four variables.
I hope someone can clarify my mishap in reading functions properly or provide me with a source where I can read all various ways functions are written.
Thank You...

EDIT: Reason why I raised this question.
I have a function z=ln(xy^2).
1. First consideration . . .
If z=f(x,y) then : F(x,y,z)=f(x,y)-z
Hence grad(F)=<1/x,2/y,-1> . . . At point(1,1,0) . . . grad(F)=<1,2,-1> . . . where grad(F) is a vector normal to the surface at that point
2. Second consideration . . .
If z=f(x,y) then I can also do . . grad(z)=<1/x,2/y> . . . At point(1,1,0) . . . grad(F)=<1,2> . . . which is also normal to the surface with a rising rate of modulus(<1,2>)

So I interpreted the problem in two ways and I had two seemingly similar representations of normal vectors to the curve with one having an extra term that I cannot interpret.
 
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  • #2
medwatt said:
So z=f(x,y) ... is a function of three variables with x and y being independent

Nope, it's an equation in 3 variables. We really don't care too much about what's a function of what here.

u=F(x,y,z)=f(x,y)-z . . . all of a sudden it seems as though z has suddenly become independent and there are now four variables.

Some clearing up should be done by explaining what exactly we mean by a graph. A graph of some equation (not always z=f(x,y) for some f) is the set of all points (x,y,z) that satisfy the equation. Once we clear that up, it shouldn't really matter what's dependent on what. It's just convention that we usually try to write z=f(x,y).

EDIT: Reason why I raised this question.
I have a function z=ln(xy^2).
1. First consideration . . .
If z=f(x,y) then : F(x,y,z)=f(x,y)-z
Hence grad(F)=<1/x,2/y,-1> . . . At point(1,1,0) . . . grad(F)=<1,2,-1> . . . where grad(F) is a vector normal to the surface at that point
2. Second consideration . . .
If z=f(x,y) then I can also do . . grad(z)=<1/x,2/y> . . . At point(1,1,0) . . . grad(F)=<1,2> . . . which is also normal to the surface with a rising rate of modulus(<1,2>)

The second is [itex]\nabla f[/itex] in two-dimensional space. It's normal to the curve [itex]f\left(x,y\right)=C[/itex], where C is a constant, not to the surface. The first is [itex]\nabla\left(f\left(x,y\right)-z\right)[/itex], where z is some new variable, independent of x and y (unless we're only looking at the gradient when we're on the surface,) which is our third variable if we're taking the gradient in three-dimensional space.
 
  • #3
If ∇f is normal to a curve and the curve lies on the surface . . . shouldn't that mean the vector is also normal to the surface because after all a vector only acts at a point ?
 
  • #4
medwatt said:
If ∇f is normal to a curve and the curve lies on the surface . . . shouldn't that mean the vector is also normal to the surface because after all a vector only acts at a point ?

Only if the surface is vertical, for instance, we could consider the plane [itex]x=z[/itex]. For given z, the gradient of the function f, where [itex]z=f\left(x,y\right)[/itex], is [itex]\left\langle1,0\right\rangle[/itex], which sticks out from the plane, but not perpendicular to it.

Another thing to note is that [itex]\left\langle1,0\right\rangle[/itex] doesn't really mean anything in a three-dimensional example. So, for this to make sense, we'd "cast" it (to use the programming term) to [itex]\left\langle1,0,0\right\rangle[/itex] for three-dimensional problems. Note that this is just an arbitrary choice of "cast" that's the most intuitive.
 

1. What is a multivariable function?

A multivariable function is a type of mathematical function that takes in more than one input variable and produces an output value. It is often used to model relationships between multiple factors or variables.

2. What is the purpose of using notations in multivariable functions?

The purpose of using notations in multivariable functions is to represent and describe the relationships between the input variables and the output value in a concise and organized manner. It allows for easier understanding and manipulation of the function.

3. What are the common notations used in multivariable functions?

The most common notations used in multivariable functions are f(x, y), g(x, y), or h(x, y, z), where the letters within the parentheses represent the input variables and the letters outside the parentheses represent the output value. Other notations include using subscripts or superscripts to denote the variables, such as fxy or h3.

4. How are multivariable functions graphed?

Multivariable functions can be graphed in a three-dimensional coordinate system, where the input variables are represented on the x and y axes and the output value is represented on the z axis. The graph will show how the output value changes in relation to the input variables.

5. Can multivariable functions have more than two input variables?

Yes, multivariable functions can have any number of input variables. The number of input variables determines the dimensionality of the function. For example, a function with three input variables would be graphed in a four-dimensional coordinate system.

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