As I understand a surface is any function in three dimensions, or a function that contains 3 variables (e.g. f(x,y,z) = x + y + z). However my math teacher gave this example:

I am wondering if the given($\displaystyle A = x^2 + y^2 + z^2 = 9$ and $\displaystyle B = z = x^2 + y^2 - 3$), really can be considered surfaces even though they are equations, and if the example is correct...Find the angle between the surfaces:

$\displaystyle A = x^2 + y^2 + z^2 = 9$

$\displaystyle B = z = x^2 + y^2 - 3$

at the point P(2,-1,2)

gradient of A = 2xi + 2yj +2zk

gradient of B = 2xi + 2yj - k

at the point P:

gradient of A = 4i - 2j + 4k

gradient of B = 4i - 2j - k

gradA dotted with gradB = $\displaystyle |{gradA}| |{gradB}| cos(theta)$

then theta = 54.4 degrees