# Thread: integral of a conservative vector field

1. ## 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).

2. 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) ....

3. 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) $\displaystyle \int_S {\bf{F}} \cdot {\bf{dr}}=\int_{t=0}^1 [(t+t^3){\bf{i}} + t^3 {\bf{j}} + (t+t^2){\bf{k}}] \cdot [{\bf{i}} +2 t {\bf{j}}+ 3 t^2 {\bf{k}}] \ dt$

RonL

4. 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.

5. Originally Posted by wik_chick88
i am completely stuck.

for (b): i told you how to find a potential function $\displaystyle f(x,y,z)$ 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 $\displaystyle C$ is a smooth curve given by the vector function $\displaystyle \bold{r}(t)$ for $\displaystyle a \le t \le b$, and $\displaystyle f$ is a continuous function whose gradient vector $\displaystyle \nabla f$ is continuous on $\displaystyle C$, then

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

note here that your $\displaystyle \bold{F} = \nabla f$ that is mentioned

6. Originally Posted by Jhevon

for (b): i told you how to find a potential function $\displaystyle f(x,y,z)$ 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 $\displaystyle C$ is a smooth curve given by the vector function $\displaystyle \bold{r}(t)$ for $\displaystyle a \le t \le b$, and $\displaystyle f$ is a continuous function whose gradient vector $\displaystyle \nabla f$ is continuous on $\displaystyle C$, then

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

note here that your $\displaystyle \bold{F} = \nabla f$ that is mentioned
ok i got for b:
$\displaystyle f(x,y,z)$ = x^2/2 + zy + xz is that right? and then how do i find
$\displaystyle \int_C \nabla f \cdot d \bold{r} = f(\bold{r}(b)) - f(\bold{r}(a))$ ?????

7. Originally Posted by wik_chick88
ok i got for b:
$\displaystyle f(x,y,z)$ = 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
$\displaystyle \int_C \nabla f \cdot d \bold{r} = f(\bold{r}(b)) - f(\bold{r}(a))$ ?????
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

8. 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?

9. 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)

10. 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)

11. 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 ....