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Thread: Gradient of scalar fields

  1. #1 Gradient of scalar fields 
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    I feel like Ive been banging my head against the wall for three days now in which Ive been trying to understand the process of creating a vector field which depicts the gradient of a scalar field.

    I understand the math but I just cant get it to "click" in my head, especially since Im having a hard time imagining this thing for a field Z(x,y,z), how the hell do you do that?

    Just to see if Im clear on this, the gradient of the field is a vector field which tells me the the rate of change of the scalar field(may I call it the "primitive", like for curves?). This gradient is denoted with the "del" operator.

    The construct that I think with is like a three dimesional coordinate system with a sperical form in it with the vectors of the gradient pointing towards the center of the sphere, the sphere being something like a star or something that gets progressivley denser as you move closer to its center, with this desity being Z(x,y,z)

    Can anybody give me a hand with this? Some form of help visualizing it or ANYTHING at all, Im studying this using Feynmans lectures.


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  3. #2  
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    Another question, am I slow for not understanding this fast enough, how long did you take to get it?


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  4. #3  
    Forum Radioactive Isotope mitchellmckain's Avatar
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    I don't understand your talk of a sphere at all.

    Here is a different example. Take an old black and white photo. An example of a scalar field would come from assigning a numerical value of the brightness (black being 0 and white being 1) at every point on the photo. The gradient of this scalar field now assigns a vector to every point on the photo which points in the direction where there is the fastest increase in brightness, and the magnitude of these vectors is the rate of increase in brightness in that direction. At points where the color is black or white, the gradient is zero and these are minima and maxima of the scalar field. You can have local minima at a gray point which is surrounded by brighter areas all around it and a local maxima at a gray spot wihich is surrounded by darker areas. There may also be a point where the gradient is zero even though it is not a minima or maxima of any kind in the scalar field, such as a saddle point, where it gets darker in two directions and brighter in two other directions.

    P.S. This is of course an example of a scalar field and its gradient in only two dimensions F(x,y).
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  5. #4  
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    I understand that what you said, but the transition to a function f(x,y,z) is what boggles me.

    The sphere, think of your black and white photo, but instead of it being just a plane, photograph, it is in space, 3d, with the black maximum in the center and its darkness constantly decreasing in every direction, spherically.
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  6. #5  
    Forum Radioactive Isotope mitchellmckain's Avatar
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    So with a spherically symmetric scalar field you get a spherically symmetric gradient with zero in the center and vectors pointing radially outward with magnitudes equal to the rate of increase in brightness.

    There are many examples of this in physics like a gravitational (force) field, which is the gradient of a scalar field called the gravitational potential (gravitational potential energy).
    See my physics of spaceflight simulator at http://www.relspace.astahost.com

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  7. #6  
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    Yeah I think I understand it now, Ive moved on to divergence and curl.
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