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Calculus questions

  1. Oct 9, 2005 #1
    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
  2. jcsd
  3. Oct 9, 2005 #2

    Tom Mattson

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    Staff Emeritus
    Science Advisor
    Gold Member

    Yes, it does.

    For the next problem, you are being asked to show that the level surfaces of [itex]f[/itex] are the spheres [itex]S: x^2+y^2+z^2=r^2[/itex]. That means that [itex]\hat{\nabla}f[/itex] must be normal to [itex]f[/itex] (why?). That should be easy enough to show.
  4. Oct 9, 2005 #3
    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?
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