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\$.

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

Jacob BachJacob Bach
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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 df xcdotf 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 df xcdotf F = -int_gamma m df xcdot ablaphi = -int_f a^f b m dphi= phi(f a) - phi(f b) ag2\$\$

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answered may 15 "18 in ~ 13:18

caveraccaverac
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