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Math Help - Physical Significance of d(f(x,y))

  1. #1
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    Physical Significance of d(f(x,y))

    What is the physical meaning of d(x2y2)?

    For example,
    d(x2) = 2x dx
    This can be derived as,

    d(x2)/dx = 2x
    d(x2) = 2x dx

    Here dx can be thought of Delta x, rather than thinking d/dx as an operation. So, small change in x2 is the product of "two times current value of x" and small change in x.

    What will be the similar deduction methodology for d(x2y2) = 2xy2dx + 2yx2dy?

    Thanks a lot.
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  2. #2
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    Re: Physical Significance of d(f(x,y))

    Given f(x,y)= x^2y^2 then grad f= \nabla f= 2xy^2\vec{i}+ 2x^2y\vec{j} is a vector pointing in the direction of fastest increase of f with length equal to the rate of change of f in that direction. We can think of " dx\vec{i}+ dy\vec{j}" as a vector a "small" distance, dx, parallel to the x-direction, and a "small distance", dy, parallel to the y-axis. So df= 2xy^2dx+ 2x^2ydy is the dot product of those two vectors, the rate of change of f along this "infinitesmal" vector.
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  3. #3
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    Re: Physical Significance of d(f(x,y))

    Thanks a lot for your reply.
    Just extending your view, and putting in my thoughts, we can think as , where z is the third axiz in the 3D- Plane. So the differential equation corresponds to the tangential plane at any point . Then, it can be seen as "the rate of change of f along this "infinitesimal" vector", just as we could map a tangent in a 2-D curve with rate of change of y with respect to x.

    Thanks a lot @HallsofIvy. Your reply was extremely helpful.
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