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Math Help - integral of a conservative vector field

  1. #1
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    integral of a conservative vector field

    The vector field F = (x + z)i + zj + (x = y)k is conservative.
    (a) Use the definition of the line integral to evaluate F ∙ dr along the path r(t) = ti + t2j + t3k for 0 t 1. The goal here is to carry out the full calculation, rather than using the fundamental theorem for line integrals).
    (b) Find a corresponding potential function f(x,y,z) such that f = F.
    (c) Use the result calculated in (b) to re-evaluate the integral in (a).
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    Quote Originally Posted by wik_chick88 View Post
    The vector field F = (x + z)i + zj + (x = y)k is conservative.
    (a) Use the definition of the line integral to evaluate F ∙ dr along the path r(t) = ti + t2j + t3k for 0 t 1. The goal here is to carry out the full calculation, rather than using the fundamental theorem for line integrals).
    (b) Find a corresponding potential function f(x,y,z) such that f = F.
    (c) Use the result calculated in (b) to re-evaluate the integral in (a).
    It would help a lot if you showed your working. Where are you stuck?

    For (c): t = 0 => (0, 0, 0) and t = 1 => (1, 1, 1). You should know that the answer to (a) will therefore be f(1, 1, 1) - f(0, 0, 0) ....
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    Quote Originally Posted by wik_chick88 View Post
    The vector field F = (x + z)i + zj + (x = y)k is conservative.
    (a) Use the definition of the line integral to evaluate F ∙ dr along the path r(t) = ti + t2j + t3k for 0 t 1. The goal here is to carry out the full calculation, rather than using the fundamental theorem for line integrals).
    (b) Find a corresponding potential function f(x,y,z) such that f = F.
    (c) Use the result calculated in (b) to re-evaluate the integral in (a).
    (a)  <br />
\int_S {\bf{F}} \cdot {\bf{dr}}=\int_{t=0}^1 [(t+t^3){\bf{i}} + t^3 {\bf{j}} + (t+t^2){\bf{k}}] \cdot [{\bf{i}} +2 t {\bf{j}}+ 3 t^2 {\bf{k}}] \ dt<br />

    RonL
    Last edited by CaptainBlack; September 11th 2008 at 01:01 AM.
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    Quote Originally Posted by wik_chick88 View Post
    The vector field F = (x + z)i + zj + (x + y)k is conservative.
    (a) Use the definition of the line integral to evaluate F ∙ dr along the path r(t) = ti + t^2j + t^3k for 0 t 1. The goal here is to carry out the full calculation, rather than using the fundamental theorem for line integrals).
    (b) Find a corresponding potential function f(x,y,z) such that f = F.
    (c) Use the result calculated in (b) to re-evaluate the integral in (a).
    i am completely stuck.
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    Quote Originally Posted by wik_chick88 View Post
    i am completely stuck.
    CaptainBlack addressed (a)

    for (b): i told you how to find a potential function f(x,y,z) before. i gave you a full solution to one of your problems here. please review it

    for (c): recall the fundamental theorem for line integrals.

    if C is a smooth curve given by the vector function \bold{r}(t) for a \le t \le b, and f is a continuous function whose gradient vector \nabla f is continuous on C, then

    \int_C \nabla f \cdot d \bold{r} = f(\bold{r}(b)) - f(\bold{r}(a))


    note here that your \bold{F} = \nabla f that is mentioned
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    Quote Originally Posted by Jhevon View Post
    CaptainBlack addressed (a)

    for (b): i told you how to find a potential function f(x,y,z) before. i gave you a full solution to one of your problems here. please review it

    for (c): recall the fundamental theorem for line integrals.

    if C is a smooth curve given by the vector function \bold{r}(t) for a \le t \le b, and f is a continuous function whose gradient vector \nabla f is continuous on C, then

    \int_C \nabla f \cdot d \bold{r} = f(\bold{r}(b)) - f(\bold{r}(a))


    note here that your \bold{F} = \nabla f that is mentioned
    ok i got for b:
    f(x,y,z) = x^2/2 + zy + xz is that right? and then how do i find
    \int_C \nabla f \cdot d \bold{r} = f(\bold{r}(b)) - f(\bold{r}(a)) ?????
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by wik_chick88 View Post
    ok i got for b:
    f(x,y,z) = x^2/2 + zy + xz is that right?
    i don't know. you have a typo in your original question. in the kth coordinate, should it be x - y or x + y?

    and then how do i find
    \int_C \nabla f \cdot d \bold{r} = f(\bold{r}(b)) - f(\bold{r}(a)) ?????
    this is just like the fundamental theorem of calculus. you know r(t), so find r(0) and find r(1) and plug it into f (which we have yet to determine). you take the x-component of the vector and put that for x in your function, the y-component for y etc
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    Quote Originally Posted by Jhevon View Post
    i don't know. you have a typo in your original question. in the kth coordinate, should it be x - y or x + y?

    this is just like the fundamental theorem of calculus. you know r(t), so find r(0) and find r(1) and plug it into f (which we have yet to determine). you take the x-component of the vector and put that for x in your function, the y-component for y etc
    sorry the original question had x+y as the component for k. so r(0) = 0i + 0j + 0k and r(1) = i + j + k. where do i go from here?
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by wik_chick88 View Post
    sorry the original question had x+y as the component for k. so r(0) = 0i + 0j + 0k and r(1) = i + j + k. where do i go from here?
    so f(r(1)) = f(1,1,1) and f(r(0)) = f(0,0,0)
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    is up to his old tricks again! Jhevon's Avatar
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    by the way, your f is right. you left of the arbitrary constant though

    it won't matter when you are applying the fundamental theorem, but it matters for your answer in part (b)
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  11. #11
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    Quote Originally Posted by Jhevon View Post
    so f(r(1)) = f(1,1,1) and f(r(0)) = f(0,0,0)
    A lot of time could have been saved if the OP re-read post #2 ....
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