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).
CaptainBlack addressed (a)
for (b): i told you how to find a potential function 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 is a smooth curve given by the vector function for , and is a continuous function whose gradient vector is continuous on , then
note here that your that is mentioned
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 etcand then how do i find
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