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Multivariable function notations 
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#1
Jan1613, 01:29 PM

P: 115

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. 


#2
Jan1613, 03:27 PM

P: 643




#3
Jan1613, 07:39 PM

P: 115

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
Jan1613, 07:46 PM

P: 643

Multivariable function notations
Another thing to note is that [itex]\left\langle1,0\right\rangle[/itex] doesn't really mean anything in a threedimensional 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 threedimensional problems. Note that this is just an arbitrary choice of "cast" that's the most intuitive. 


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