what are level curves? please explain in easy way..
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Level curves are explained at the beginning of this lecture:
f(x,y)=c is a level curve what do you think about f(x,y)=z and f(x,y,z)=0
are these also level curves? if so then please tell the reason if you can. also tell me that in level curves the space between one level curve to another must be same?
what's the difference between level curves and level surfaces?
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i have listened the lecturere and found it very helpful but i want to clearify above points. please explain if u can...
z = f(x,y) is a way to graph the function f(x,y) in the 3-dim space. In 3-d space,
z = f(x,y) describes a surface. The level curves in this case are values where f(x,y) = c, a constant. Equivalently, it's when z = c, because z = f(x,y).
If f(x,y) = 0, that will describe the level curve for when the function is zero. Equivalently, it describes the level curve for when z = 0 where z = f(x,y).
f(x,y,z) is a function of three variables. ideally, we would graph it in 4-d space. But we have a very hard time imagine 4-d space. For that reason, we consider the equivalent of level curves for this function: f(x,y,z) = c. But now, instead of describing curves in 2-d space as do level curves of the form f(x,y) = c, the equation f(x,y,z) = c describes a surface. For example, if the function where
\[ f(x,y,z) = x^2 + y^2 + z^2 \]
then f(x,y,z) = c, where c > 0, describes a sphere.