Question

Why does z+1=x*e^y*cosz can be rewritten to F(x,y,z)=-z+x*e^y*cosz?

Answer 1

the 1 just disappears like that? so if there is a 1 you can rewrite it like that?

Answer 2

Not in general, no. But what does it mean given your function F if F(x,y,z) = 1 ?

Answer 3

sorry I don't understand your question?

Answer 4

unit vector?

Answer 5

ohh does it tell the shape it is suppose to be

Answer 6

F(x,y,z) is a function from R^3 to R. F(x,y,z) = c, a constant, is a level set of the function.

In particular, if F(x,y,z)=-z+x*e^y*cosz,
then if F(x,y,z) = 1 then
-z+x*e^y*cosz = 1
i.e.,
x*e^y*cosz = 1 + z

Therefore this last equation describes the set in R^3 such that F(x,y,z) = 1.

Answer 7

Thanks!

Answer 8

Another example. Consider the function \( g : \mathbb{R}^2 \to \mathbb{R} \) defined by
\[ g(x,y) = x^2 -2x + y^2 \]
Then the level sets of \( g \) are circles: e.g., g(x,y) = 5 iff \( (x-1)^2 + y^2 = 2^2 \).