- #1

medwatt

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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.

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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