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Math Help - Vector Field

  1. #1
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    Vector Field

    If

    F(x,y,z) = sinyi + (xcosy + z )j + yk

    find a function f (x,y,z) such that

    F = grad f

    how would you go about solving this one?
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Quote Originally Posted by heatly View Post
    If

    F(x,y,z) = sinyi + (xcosy + z )j + yk

    find a function f (x,y,z) such that

    F = grad f

    how would you go about solving this one?
    \frac{\partial f}{\partial x}=sin y

    f=xsin y+A(y,z)

    \frac{\partial f}{\partial y}=xcos y+z

    f=xsin y+yz+B(x,z)

    \frac{\partial f}{\partial z}=y

    f=yz+C(x,y)

    By inspection, we get f=xsin y+yz
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  3. #3
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    Perhaps a touch simpler: from f_x= sin(y) we get f= xsin(y)+ A(y,z)

    Now differentiate that with respect to y: f_y= xcos(y)+ A_y= x cos(y)+ z so that A_y= z. That tells us that A(y,z)= yz+ B(z). f(x,y,z)= x sin(y)+ A(y,z)= x sin(y)+ yz+ B(z).

    Differentiating with respect to z, f_z= y+ B'(z)= y so B'= 0 and B is a constant.

    f(x,y,z)= x sin(y)+ yz+ B

    Alex Mahone left off the constant because the problem only asked for "a" function having that gradient.
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  4. #4
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    Let me add one:
    let \omega=f_xdx+f_ydy+f_zdz. It's easy to verify that d\omega=0. So F is really conservative and the line integral of \omega doesn't depend on the choice of path.
    For any point p=(x_0, y_0, z_0), Choose a path C with the following 3 segments:
    I: from (0,0,0) to (0, y_0, 0), along the y-axis
    II: from (0,y_0, 0) to (0, y_0, z_0), along the direction of z-axis
    III: from (0, y_0, z_0) to (x_0,y_0,z_0), along the direction of x-axis
    The f(p)=f(0)+\int_C \omega = f(0)+\int_I 0 dy + \int_{0}^{z_0} y_0 dz + \int_{0}^{x_0} sin(y_0)dx
    =y_0 z_0 + sin(y_0) x_0 + f(0)
    Last edited by xxp9; March 19th 2011 at 08:36 PM.
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  5. #5
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    Thanks everybody for your help.
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  6. #6
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    Re: Vector Field

    I'm new to the forum.
    Question: Can you please explain yow are you pasting in these nice italicized vector function equations?
    I've fooled around with typing in Word and doing print screen paste or upload a jpeg but these don't seem to work.
    Thanks.
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  7. #7
    MHF Contributor alexmahone's Avatar
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    Re: Vector Field

    Quote Originally Posted by JohnMboston View Post
    I'm new to the forum.
    Question: Can you please explain yow are you pasting in these nice italicized vector function equations?
    I've fooled around with typing in Word and doing print screen paste or upload a jpeg but these don't seem to work.
    Thanks.
    Click "Reply With Quote" to see the source code of any particular post.
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  8. #8
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    Re: Vector Field

    Quote Originally Posted by JohnMboston View Post
    I'm new to the forum.
    Question: Can you please explain yow are you pasting in these nice italicized vector function equations?
    I've fooled around with typing in Word and doing print screen paste or upload a jpeg but these don't seem to work.
    Thanks.
    there is also a application in start menu>>accessories>>math
    input panel. use tht.
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  9. #9
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    Re: Vector Field

    Thanks for the tip. The Math Input Panel works for me in, say, MS Word but I cannot make entries into a Forum message box. If I copy and paste from Word I lose things like the right-pointing vector line symbol above capital F or the hats above i+j in a vector function. Suggestions?
    (Sorry, I'm an email guy, not a whiz at online forums, but I'm learning.)
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  10. #10
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    Re: Vector Field

    correct even i tried it but doesnt work. sorry mate if i find something ill post it
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