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Math Help - Divergence

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    Divergence

    What is the Divergence? is it only the Partial derivatives?

    Lets say I have a vector field: F=x^2i+y^2j+z^2k, the divergence is F=2xi+2yj+2zk?

    And if it is, than what is the gradient?
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    Quote Originally Posted by asi123 View Post
    What is the Divergence? is it only the Partial derivatives?

    Lets say I have a vector field: F=x^2i+y^2j+z^2k, the divergence is F=2xi+2yj+2zk? Mr F says: Yes.

    And if it is, than what is the gradient?
    You can only take the gradient of a scalar field. You can take the curl of F if you want .....
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by asi123 View Post
    What is the Divergence? is it only the Partial derivatives?

    Lets say I have a vector field: F=x^2i+y^2j+z^2k, the divergence is F=2xi+2yj+2zk?

    And if it is, than what is the gradient?


    Does this clarify things?

    --Chris
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  4. #4
    Rhymes with Orange Chris L T521's Avatar
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    Originally Posted by asi123
    What is the Divergence? is it only the Partial derivatives?

    Lets say I have a vector field: F=x^2i+y^2j+z^2k, the divergence is F=2xi+2yj+2zk? Mr F says: Yes. Chris says: Not quite. This would be your gradient, but not your divergence...
    Mr. F, when you take the divergence of a field, you're left with a scalar value...it should be 2x+2y+2z, not 2xi+2yj+2zk...

    --Chris
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    Quote Originally Posted by Chris L T521 View Post


    Does this clarify things?

    --Chris

    So one is like the dot product and the other is the cross product?

    But still, isnt the gradient already define the partial derivative? why do I need the Divergence?
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    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by asi123 View Post
    So one is like the dot product and the other is the cross product?
    This is the proper way of writing it in vector notation. Yes, one is a dot product, and the other is a cross product. Divergence and Curl are important in the study of fluids [this is where the names originated]. Divergence tells you how the fluid flows towards or away from a point; Curl tells you the rotational properties of the fluid. It would make sense that divergence is a scalar, and curl would be a vector.

    But still, isnt the gradient already define the partial derivative?
    Yes, the gradient consists of partial derivatives, but note the difference between the values of the gradient and divergence:



    Does this make a little more sense now?

    --Chris

    EDIT: I should have made this a little clearer by noting that the gradient can only be applied to scalar fields, as Mr. Fantastic had mentioned earlier.
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    Quote Originally Posted by Chris L T521 View Post
    This is the proper way of writing it in vector notation. Yes, one is a dot product, and the other is a cross product. Divergence and Curl are important in the study of fluids [this is where the names originated]. Divergence tells you how the fluid flows towards or away from a point; Curl tells you the rotational properties of the fluid. It would make sense that divergence is a scalar, and curl would be a vector.



    Yes, the gradient consists of partial derivatives, but note the difference between the values of the gradient and divergence:



    Does this make a little more sense now?

    --Chris

    Yeah man, that makes a lot of sense.

    thanks a lot.
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    Quote Originally Posted by Chris L T521 View Post
    Mr. F, when you take the divergence of a field, you're left with a scalar value...it should be 2x+2y+2z, not 2xi+2yj+2zk...

    --Chris
    My mistake. I didn't look closely enough and so saw what I expected to see ......
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