Stuck on this problem

Homework Statement Not sure how to approach this one, does anyone have an idea?

BTW, in the top line it should read xz-plane, not xy.

tiny-tim
Homework Helper
Hi diffusion! (try using the X2 tag just above the Reply box )

Forget all the stuff about the cylinder …

the question is simply to prove that if you rotate z = e-x2 about the z-axis, you get z = e-x2-y2

that's easy, isn't it? Hi diffusion! (try using the X2 tag just above the Reply box )

Forget all the stuff about the cylinder …

the question is simply to prove that if you rotate z = e-x2 about the z-axis, you get z = e-x2-y2

that's easy, isn't it? I'm sure it's easy once I've been set in the right direction. I can see that z = e-(x2-y2) by graphing it, I just don't know how to show mathematically that it does.

tiny-tim
Homework Helper
ok … what does "rotate z = f(x) about the z-axis" mean? … how would you express the instruction as an equation? ok … what does "rotate z = f(x) about the z-axis" mean? … how would you express the instruction as an equation? Well you will rotate the function around the z-axis, so that z = f(x) becomes z = f(x,y). Not sure how to express it as an equation, besides telling you that the equation is a function of both x and y.

tiny-tim
Homework Helper
Try putting it in words first …

what happens to each individual point on the original curve?​

Try putting it in words first …

what happens to each individual point on the original curve?​

Produces a circle when rotated.

tiny-tim
Homework Helper
Produces a circle when rotated.

Yup! And the equation of a circle is … ? Yup! And the equation of a circle is … ? Yep, I recognized this before I posted the question. Again I just don't know how to show it, or justify it. I suppose I would say something like

"Since each point on the graph z = e-r2 will generate a circle when rotated about the z-axis, and the equation for a circle is r2 = x2 + y2, we can make this substitution into our equation for r2, giving us z = e-(x2+y2)."

I don't know, somehow this justification seems vague and inadequate.

tiny-tim
Homework Helper
"Since each point on the graph z = e-r2 will generate a circle when rotated about the z-axis, and the equation for a circle is r2 = x2 + y2, we can make this substitution into our equation for r2, giving us z = e-(x2+y2)."

ok, let's formalise that …

the point (x0,0,z0) generates the circle (x,y,z), where … ? ok, let's formalise that …

the point (x0,0,z0) generates the circle (x,y,z), where … ? At z0? Sorry, not sure if I really understand what you're asking. Are you asking for a point, the equation of the particular circle generated, or something else?

tiny-tim
I'm asking for the x,y,z equation(s) of the circle generated by the initial point (x0,0,z0). 