Fundamental theorem of line integrals

Question

F=e^y i+(xe^y+e^z)j + (ye^z)k evaluate ⌠Fdr along A(0,2,0) and B(4,0,3). Find if its conservative and if so find its potential then use A and B. 

Answer 1

If F = ∇U then we find that:

 

U = xe^y + f(y,z)

 

U = xe^y + ye^z + g(x,z)

 

U = ye^z + h(x,y)

 

We can take:

 

f(y,z) = ye^z

 

g(x,z) = 0

 

h(x,y) = xe^y

 

This means that F(x,y,z) is conservative with potential:

 

U(x,y,z) = xe^y + ye^z

 

Thus ∫_A^B F·dr = U(B) - U(A) = (4e⁰ + 0e³) - (0e² + 2e⁰) = 4 - 2 = 2

 

This is independent of the path taken.