i am brand-new to vector calculus and also found this problem in the textbook which ns am not sure how to work with, yet I want to learn to perform so:

Show details to find the occupational done through the force field $ hedesigningfairy.combfF =x^3, hedesigningfairy.combfi+y^3, hedesigningfairy.combfj$ in moving an item from $P(1, 0)$ to $Q(2, 2).$



In this case, you can write your pressure as a gradient of a potential $phi$,

$ hedesigningfairy.combfF=- ablaphi$,

where $phi$ is her potential,

$phi=-frac14 (x^4+y^4)$.

You are watching: Find the work done by the force field f in moving an object from p to q.

You can check that this offers you the right force since $F_i=-fracdphid x=x^3$ and $F_j=-fracdphid y=y^3$.

As shortly as you have a potential, the job-related $W_ extdone by force$ done by the pressure on any route from $P$ come $Q$ is just the distinction in the potential,

$W_ extdone by force=phi_start-phi_end=phi(P)-phi(Q) =(-frac14(1^4+0^4))-(-frac14(2^4+2^4))=frac314$.

edited might 15 "18 in ~ 14:31
answered might 15 "18 in ~ 13:13

Jacob BachJacob Bach
19088 bronze title
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Just desire to complement
JacobBach"s answer, the work done for the pressure $f F$ ~ above the route $gamma$ is calculated together

$$W = int_gamma m df xcdotf F ag1$$

If $f F$ deserve to be written as together the gradient that a potential field

$$f F = - abla phi$$

then (1) becomes

$$W = int_gamma m df xcdotf F = -int_gamma m df xcdot ablaphi = -int_f a^f b m dphi= phi(f a) - phi(f b) ag2$$

point out
answered may 15 "18 in ~ 13:18

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