Calculus Questions: Help Solving Problems

[Pearce_09]

Hello there, I have a few problems with a recent calculus assignment and i was wondering if someone could give me a hand.

1. Suppose that the temperature at the point (x,y,z) in space is T(x,y,z) = x^2 + y^2 + z^2. Let a particle follow the right-circular helix sigma(t) = (cost, sint, t) and let T(t) be its temperature at time t,
a) What is T'(t)

I thought that since T is the temperature at a point, but the particle travles along the helix i could write
T(t) = (costt, sint, t) = (cost)^2 + (sint)^2 + t^t = 1 + t^2
then T'(t) = 2t

does that make sense?

My next problem i am quite stumped on, and even a slight tap in the general right direction would be much appreciated.
2. Let f and g be functions R^3=>R. Suppose f is differentiable and

(Gradient)f = (partial f wrt x, partial f wrt y, partial f wrt z) = g(x,y,z)(x,y,z)

Show that f is constant on any sphere radius r centered at the origin defined by x^2 + y^2 + z^2 = r^2.

Thanks alot

 

[quantumdude]

I thought that since T is the temperature at a point, but the particle travles along the helix i could write
T(t) = (costt, sint, t) = (cost)^2 + (sint)^2 + t^t = 1 + t^2
then T'(t) = 2t

does that make sense?

Yes, it does.

For the next problem, you are being asked to show that the level surfaces of f are the spheres S: x^2+y^2+z^2=r^2. That means that \hat{\nabla}f must be normal to f (why?). That should be easy enough to show.

 

[Pearce_09]

Thank you, i wasn't sure if i had done that one correctly.

As for question two, that definatly clears up some things, however I'm having trouble showing this.
If F is constant, the gradient of F is normal. So to show that F is constant on any sphere defined by S: x^2 + y^2 + z^2 = r^2 i have to show that the gradient of F is normal, that is the inner product of the gradient of F with the tangent vector v is zero.. right? however, I'm not quite sure how to do that.. what is the purpose of g(x,y,z)(x,y,z) in the question?