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Math Help - tangent line integral

  1. #1
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    tangent line integral

    I can't seem to get a ''nice'' solution to this question, the answer should be a simple fraction not what i get.
    The question is 'calculate the line integral of the vector field
    v(x,y,z)=((x-1)(z-3),xyz,x+z) along the straight line from (1,1,1) to (1,3,9)

    I started by parameterising the line:
    d(t)=(1,1,1)+t(0,2,8)
    d(t)=(1,1+2t,1+8t)
    d'(t)=(0,2,8)

    thus v(d(t))=(0,(1+2t)(1+8t),(2+8t)

    then i go to work out the integral between 0 and 1 of
    <(v(d(t)),d'(t)> dt
    and it comes out crap.
    Any help would be good, pointing our error or next step
    thanks
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by BlackadderVII View Post
    I can't seem to get a ''nice'' solution to this question, the answer should be a simple fraction not what i get.
    The question is 'calculate the line integral of the vector field
    v(x,y,z)=((x-1)(z-3),xyz,x+z) along the straight line from (1,1,1) to (1,3,9)

    I started by parameterising the line:
    d(t)=(1,1,1)+t(0,2,8)
    d(t)=(1,1+2t,1+8t)
    d'(t)=(0,2,8)

    thus v(d(t))=(0,(1+2t)(1+8t),(2+8t)

    then i go to work out the integral between 0 and 1 of
    <(v(d(t)),d'(t)> dt
    and it comes out crap.
    Any help would be good, pointing our error or next step
    thanks
    So you want \displaystyle \int_C \bold F \cdot d \bold r, where \displaystyle C is the straight line segment from (1,1,1) to (1,3,9)

    You started fine. you would end up with:

    \displaystyle \int_0^1 \left< 0, (1 + 2t)(1 + 8t), 2 + 8t \right> \cdot \left< 0, 2, 8 \right> ~dt = \int_0^1 \left( 32t^2 + 84t + 18 \right)~dt

    You said you did this and it was wrong?
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  3. #3
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    why do you seperate <(0,(1+2t)(1+8t),(2+8t) , (0,2,8)>?
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by BlackadderVII View Post
    why do you seperate <(0,(1+2t)(1+8t),(2+8t) , (0,2,8)>?
    what do you mean? it is F * dr, that's a dot product of two vectors. so that's what i did. F = v is one vector, and dr is another. you take their dot product and integrate...
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  5. #5
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    Quote Originally Posted by Jhevon View Post
    \displaystyle \int_0^1 \left< 0, (1 + 2t)(1 + 8t), 2 + 8t \right> \cdot \left< 0, 2, 8 \right> ~dt = \int_0^1 \left( 32t^2 + 84t + 18 \right)~dt
    Could you possible explain this step?

    edit: ignore me, i've just realised my mistake. thanks a lot Jhevon
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