Question

Consider F and C below.F(x, y, z) = yzexzi + exzj + xyexzk,C: r(t) = (t2 + 5)i + (t2 − 1)j + (t2 − 5t)k, 0 ≤ t ≤ 5(a) Find a function f such that F = ∇f.f(x, y, z) = (b) Use part (a) to evaluate CF · dr along the given curve C.

Answer 1

`(\partial f)/(\partial x)=yze^(xz)\implies f(x,y,z)=ye^(xz)+g(y,z)`

`(\partial f)/(\partial y)=e^(xz)=e^(xz)+(\partial g)/(\partial y)\implies(\partial g)/(\partial y)=0\implies g(y,z)=h(z)`

`(\partial f)/(\partial z)=xye^(xz)=xye^(xz)+(\mathrm dh)/(\mathrm dz)\implies(\mathrm dh)/(\mathrm dz)=0\implies h(z)=C`

`\implies f(x,y,z)=ye^(xz)+C`

Then the value of any integral of
`\vec F` along
`C` is
`f(x_2,y_2,z_2)-f(x_1,y_1,z_1)`, where
`(x_1,y_1,z_1)` and
`(x_2,y_2,z_2)` are the endpoints of the path
`C`.