some questions on sketching level sets

Question

I have two questions on level sets :

1) given a 3-dimensional function such as $f(x,y,z)=x^2+y^2+z^2$ and I want to find the level sets at c=0,1,-1. Then do I draw a 3-dimensional image ?

I can imagine that 1 gives a sphere , 0 gives the shape seen in nuclear plants ( cant remember what its called) and Im not sure on what -1 gives though I feel it will be a 3-d hyperbola type shape.

2) How do we use level sets of a function to sketch pictures of the gradient vector field ? ? say to calculate the gradient vector of $\phi(x,y)=y^2-x^2$. do we take it that $grad\phi=-2xi+2yj$. which means that at the point (1,0) we'll have a vector pointing from (1,0) to (-1,0) etc....( do we only do this along the curve or over the whole plane ?

Answer 1

Hint: $x^{2}+y^{2}+z^{2}=-1$ describes the sphere with radius $i$, and you are correct in saying that $c=0$ gives the nuclear plant shape, but note that it is around the $z$-axis.

Answer 2

Hint1: What shape does $x^2+y^2+z^2=r^2$ describe?

It is always a sphere

Hint2: What does the gradient physically mean? there is no need to calculate the gradient along the level sets to get the gradient field

How the level sets are changing