Painguy said:Homework Statement
given f(x,y)=9/(x+y) find a level surface.
Homework Equations
The Attempt at a Solution
g(x,y,z)=f(x,y)-z=0?
g(x,y,z)=9/(x+y) -z=0?
That answer is wrong. Apparently i must have the following:
g(x,y,z)=z(x+y)=9
How do I solve problems like these?
Another example is to find a function f(x,y,z) whose level surface f=5 is the graph of the paraboloid
g(x,y)=-x^2 -y^2
LCKurtz said:You don't need any ##z## variable. To find level surface of a function ##f(x,y)## just plot the graphs of ##f(x,y)=C## for various constants ##C##. For a two variable problem like yours, they will be curves in the ##xy## plane, not surfaces.
Painguy said:Wouldn't that be generating level curves? I guess my terminology is bad. What i am asking is how do make f(x,y) into
F(x,y,z)
A level surface is a two-dimensional representation of a three-dimensional function. It is created by plotting points on a graph where the function has the same output value, resulting in a surface that appears flat.
A level surface can be generated by setting the output value of the function to a constant and solving for one of the input variables. This will result in a graph of points representing all the possible combinations of the remaining input variable that produce the given output value.
Generating a level surface allows for a visual representation of a three-dimensional function, making it easier to analyze and understand its behavior. It can also help in identifying critical points, such as maxima and minima, and understanding the relationship between the input and output variables.
The shape of a level surface can be affected by the coefficients and exponents of the input variables in the function. Changing these values can result in a different shape or orientation of the level surface.
Yes, a level surface can be generated for any two-variable function, as long as the function is continuous and has a unique output value for every combination of input variables. This means that functions with singularities or undefined points cannot have a level surface.