Find the form of f(x,y) knowing the form of contour lines in the XZ and YZ planes. I am trying to find the form of z=f(x,y).

Kailyn Hamilton

Kailyn Hamilton

Answered question

2022-11-14

Find the form of f(x, y) knowing the form of contour lines in the XZ and YZ planes
I am trying to find the form of z = f ( x , y ). I know that:
Contour lines in the XZ plane are of the form:
z = A l n ( x ) + B
(the A and B parameters vary with y)
Contour lines in the YZ plane are of the form:
z = C y 2 + D y + E
(the C, D and E parameters vary with x)
What is then the form of z = f ( x , y )?

Answer & Explanation

Neil Short

Neil Short

Beginner2022-11-15Added 17 answers

Step 1
I'm not sure what you mean by A and B varying with y in the context of contour lines in the XZ plane (and likewise for x and YZ). I'll assume what you mean is that there is a function z = f ( x , y ) and the first equation describes the relationship between x and z if you hold y fixed and the second equation describes the relationship between y and z if you hold x fixed.
Step 2
Since the two expressions for z have to be equal, the dependency of the parameters on y in the first equation must be such that a quadratic function of y results for all x. Since lnx takes different values for different x, this can only happen if A and B are both quadratic functions of y. Thus, the most general form for z compatible with the given conditions is
z = ( a 2 y 2 + a 1 y + a 0 ) ln x + b 2 y 2 + b 1 y + b 0
with
A = a 2 y 2 + a 1 y + a 0 , B = b 2 y 2 + b 1 y + b 0 , C = a 2 ln x + b 2 , D = a 1 ln x + b 1 , E = a 0 ln x + b 0 .

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