integral of a conservative vector field

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September 10th 2008, 06:14 AM
wik_chick88

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 Ñf = F.
(c) Use the result calculated in (b) to re-evaluate the integral in (a).
September 10th 2008, 01:35 PM
mr fantastic

Quote:

Originally Posted by wik_chick88

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 Ñf = 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) ....
September 10th 2008, 10:53 PM
CaptainBlack

Quote:

Originally Posted by wik_chick88

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 Ñf = 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
September 14th 2008, 04:40 PM
wik_chick88

Quote:

Originally Posted by wik_chick88

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 Ñf = F.
(c) Use the result calculated in (b) to re-evaluate the integral in (a).

i am completely stuck.
September 14th 2008, 05:03 PM
Jhevon

Quote:

Originally Posted by wik_chick88

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
September 14th 2008, 07:01 PM
wik_chick88

Quote:

Originally Posted by Jhevon

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
$\int_C \nabla f \cdot d \bold{r} = f(\bold{r}(b)) - f(\bold{r}(a))$ ?????
September 14th 2008, 07:10 PM
Jhevon

Quote:

Originally Posted by wik_chick88

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?

Quote:

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
September 14th 2008, 07:16 PM
wik_chick88

Quote:

Originally Posted by Jhevon

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?
September 14th 2008, 07:49 PM
Jhevon

Quote:

Originally Posted by wik_chick88

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)
September 14th 2008, 07:57 PM
Jhevon

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)
September 14th 2008, 07:59 PM
mr fantastic

Quote:

Originally Posted by Jhevon

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 ....