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Math Help - Help with Gradient Of function problem

  1. #1
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    Help with Gradient Of function problem

    Suppose (x,y) is a point in the first quadrant (x,y>0). If delta is the angle of (x,y) in polar representation, determine the length and direction of gradient of delta(x,y)?
    can someone show me how to approach this type of question?
    Thanks for the help!
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  2. #2
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    Re: Help with Gradient Of function problem

    Quote Originally Posted by Ricebunny View Post
    the angle of (x,y) in polar representation
    ... equals arctan (y/x).

    arctan(y/x) - Wolfram|Alpha

    Notice the partial derivative wrt x. Click on 'show steps'. Just in case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    The general drift is...



    gradient of delta(x,y)?
    ... equals a vector composed of both partial derivatives:

    Gradient - Wikipedia, the free encyclopedia

    gradient of arctan(y/x) - Wolfram|Alpha

    determine the length and direction
    .. as for any vector: the length or magnitude using pythagoras and the direction relative to the x-axis using arctan as we did above for delta.

    For length, simplify \sqrt{{(\tfrac{-y}{x^2 + y^2})}^2 + {(\tfrac{x}{x^2 + y^2})}^2}

    For direction, arctan of a simplification of the 'y' component over the 'x'.



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    Balloon Calculus; standard integrals, derivatives and methods

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    Last edited by tom@ballooncalculus; June 19th 2012 at 01:33 PM.
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  3. #3
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    Re: Help with Gradient Of function problem

    Thank you so much for the help!
    Now I understand much better.
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