Question

how do I determine the level curve f(x,y)=0 when f(x,y)=(x-y)/x^2+y^2+1

Answer 1

don't you just set
(x-y)/x^2+y^2+1 = 0

and simplify ?

Answer 2

How would I determine the gradient of the same f(x,y)?

Answer 3

the gradient is a vector ( fx ,fy)
where fx means the partial derivative wrt x, and fy is the partial wrt y

Answer 4

see http://betterexplained.com/articles/vector-calculus-understanding-the-gradient/