Conservative and irrotational vectors

[Bevyclare]

1. Please can someone assist me in solving the attached vector problem.

I have made several attempt to no avail





3.

 

[vela]

You've almost got part (i) finished. You just have to keep going. You got to this point:

$$J=\int_C (ye^{xy}dx+(xe^{xy}+1)dy$$

Then you introduced a parameter u and apparently wrote x=u and y=u³ since C is the curve $y=x^3$. This gave you dx=du and dy=3u²du, so you got to here:

$$J=\int_{u_1}^{u_2} (ye^{u^4}du+3u^3e^{u^4}du+3u^2du)$$

So far, so good. You just need to substitute for y in terms of u in the first term and figure out what u₁ and u₂ are. You were told the contour runs from x=1 to x=6. Since you know how x and u are related, you can easily see what u₁ and u₂ equal.

All the terms can easily be integrated either directly or with a simple substitution

 

[Bevyclare]

Vela,
Thanks for your response.

I still cannot solve the problem.

Please let someone assist.

 

[HallsofIvy]

Where are you haveing trouble?
You already had
$$J=\int_{u_1}^{u_2} (ye^{u^4}du+3u^3e^{u^4}du+3u^2du)$$
and vela pointed out that, since $y= u^3$, that is the same as
$$J=\int_{u_1}^{u_2} (u^3e^{u^4}du+3u^3e^{u^4}du+3u^2du)$$
so it is really just
$$J=\int_{u_1}^{u_2} (4u^3e^{u^4}du+3u^2du)$$

You are told that x runs from 1 to 6. When x= 1 what is u? When x= 6, what is u?

If you still can't do it, please show your work so we can see exactly where you have a problem.

 

[Bevyclare]

Hello everyone

Please find attached the output of my integration and substitution.

However, I am not convinced that this is correct.

Please assist me.

 

[Bevyclare]

Please let someone assist me by reviewing my attached solution.

I don't think I'm correct.

Please I am counting on you.