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Math Help - Potential of a function

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    Potential of a function

    The function is given as a vector...

    F(x,y) = (x*e^((-x)^2) - (-y)^2))i + (y*e^((-x)^2 - (-y)^2))j

    Now according to the solutions provided, the potential of that function is the integral of df/dx and df/dy.

    The answer came out to (-1/2)e^((-x)^2) - (-y)^2)) + C1(y) for df/dx. I think I'm getting confused about integrating the exp, where did the x go? Thanks.
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    I understand how the potential part works, I guess I'm just confused about the integral of an exp. I don't understand how the integral of x*e^(...) goes to -(1/2)*e(...)

    Could you explain that? Thanks.
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    Quote Originally Posted by pakman View Post
    I understand how the potential part works, I guess I'm just confused about the integral of an exp. I don't understand how the integral of x*e^(...) goes to -(1/2)*e(...)

    Could you explain that? Thanks.
    Think about it in reverse. What's the derivative of e^{x^2}?

    d/dx[e^{x^2}] = e^{x^2} * (2x) = 2x*e^{x^2}

    by the chain rule. So
    Int[x*e^{x^2} dx] = (1/2)*e^{x^2} + C

    Alternately: Do a substitution.
    Int[x*e^{x^2} dx]

    Let y = x^2 ==> dy = 2x dx
    Thus
    Int[x*e^{x^2} dx] = Int[(1/2)e^y dy] = (1/2)*e^y + C = (1/2)*e^{x^2} + C

    -Dan
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