1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Nabla operator

  1. Jun 8, 2007 #1
    Could some one explain what does Nabla operator actually signify ? I understand that the various products with nabla are used to find curl,divergence,gradient in EM, but what does Nabla represent in itself ? A more basic question would be, what does del operator(partial derivative) represent , in a 3d system ?
     
  2. jcsd
  3. Jun 8, 2007 #2

    chroot

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    It's a generalization of the derivative operator to multiple dimensions. That's all.

    In one dimension, say along the x-axis, the derivative operator looks like this:

    [itex]
    \frac{d}
    {{dx}} = \frac{\partial }
    {{\partial x}} = \vec i \frac{\partial }
    {{\partial x}}
    [/itex]

    Since there's only one dimension, the "normal" derivative and partial derivative are the same. Also, there's only way way to take a derivative in one dimension -- along that dimension. Thus, the [itex]\vec i[/tex] is implied.

    In multiple dimensions, say x, y and z, it looks like:

    [itex]
    \nabla =
    \vec i \frac{\partial }
    {{\partial x}} + \vec j \frac{\partial }
    {{\partial y}} + \vec k \frac{\partial }
    {{\partial z}}
    [/itex]

    Same thing, just with more dimensions.

    - Warren
     
  4. Jun 8, 2007 #3

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    I would disagree there, chroot.
    The one-dimensional analogue of the partial derivative at a point, is the derivative with respect to the elements of some particular sequence converging to that point.

    Remember that existence of all partial derivatives does not guarantee differentiability at that point; some similar restriction ought to be provable for "sequential" derivatives in the one-dimensional case.
     
  5. Jun 8, 2007 #4

    chroot

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    So you're saying that del doesn't reduce to the standard one-dimensional derivative d/dx? Can you explain a bit more?

    - Warren
     
  6. Jun 9, 2007 #5
    What I was referring to was, what does a partial derivative represent (in multiple dimensions) like it represents slope in one dimension co-ordinate system ?
     
  7. Jun 9, 2007 #6
    a gradient vector. the direction in which the function is changing most rapidly, and the magnitude is the amount its changing
     
  8. Jun 9, 2007 #7
    If it acts on a scalar field.
     
  9. Jun 9, 2007 #8

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Hmm..what I meant is that a partial derivative is the derivative with respect to some proper subset of arguments in the vicinity of the point.
    For example along the x-axis (or some line parallell to that) in 2-D, or along the rationals in the 1-D case.
     
    Last edited: Jun 9, 2007
  10. Jun 9, 2007 #9

    cepheid

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    It's still a vector operator...so I would say that in one dimension:

    [tex] \nabla = \hat{x}\frac{d}{dx} [/tex]


    I'm sure Arildno's answer was better, but I didn't follow what he was saying.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Nabla operator
  1. Nabla Operator (Replies: 3)

  2. Vector operations (Replies: 7)

  3. Operational amplifier (Replies: 2)

  4. Operational amplifier (Replies: 13)

Loading...