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I A directional, partial derivative of a scalar product?

  1. Feb 5, 2017 #1
    Let's say I have two vector fields a(x,y,z) and b(x,y,z).

    Let's say I have a scalar field f equal to ab.

    I want to find a clean-looking, simple way to express the directional derivative of this dot product along a, considering only changes in b.

    Ideally, I would like to be able to express this without invoking unit vectors and without || 's, while still using vector notation.
     
  2. jcsd
  3. Feb 5, 2017 #2
    If vectors in a had x-components only, and vectors in b had y-components only, wouldn't ## \vec a \cdot (\nabla \vec b)## still return values with y-components only provided that b varies with x, as opposed to zero, which ab would be equal to everywhere?

    Edit: Maybe I am conflating ## \vec a \cdot (\nabla \vec b)## with ## (\vec a \cdot \nabla) \vec b##

    Edit2: Also, It seems like you gave the gradient of the dot product of ab, considering only changes in b. I was wondering about the directional derivative of ab along a, considering only changes in b.
     
    Last edited: Feb 5, 2017
  4. Feb 5, 2017 #3

    Henryk

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    Gold Member

    My mistake
     
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