Question

Please help with problem both a and b!!! For a i have xyz^2-yz+c, i do not know if this is correct and if so I do not know how to proceed with part b. Please any advise is appreciated!!

a. Find a potential function for F(x,y,z)=⟨yz^2 ,xz^2 −z,2xyz−y⟩.
b. Use part (a) to evaluate ∫c F*ds where C is any path in space from P = (1,2,3) to Q = (3,2,1).

Answer 1

F=nabla(f)
partial derivative of F w.r.t x=fx
F=F1i+F2j+F3k=fxi+fyj+fzk
fx=yz^2
so f=xyz^2+C1(y,z)
similarly
fy=xz^2+C1y=xz^2
===>
C1=C2(z)
fz=2xyz+C2'=2xyz−y
so C2=-yz+c
therfore f=xyz^2-yz+c

Answer 2

ds=sqrt(dx^2+dy^2+dz^2)????
or it is dr=dxi+dyj+dzk and operation between them is dot??

Answer 3

its #4

Answer 4

for part a, I think jamalahmed2068 showed that f = xyz^2-yz+c, thus grad(f) = F
for part b, assuming C is "smooth", then by the fundamental theorem of calculus (or line integral), ∫c F*ds = ∫c grad(f)*ds = f(Q) - f(P) = f(3,2,1) - f(1,2,3) = 4 - 12 = -8