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

$\dfrac{\partial f}{\partial x}=yze^{xz}\implies f(x,y,z)=ye^{xz}+g(y,z)$

$\dfrac{\partial f}{\partial y}=e^{xz}=e^{xz}+\dfrac{\partial g}{\partial y}\implies\dfrac{\partial g}{\partial y}=0\implies g(y,z)=h(z)$

$\dfrac{\partial f}{\partial z}=xye^{xz}=xye^{xz}+\dfrac{\mathrm dh}{\mathrm dz}\implies\dfrac{\mathrm dh}{\mathrm dz}=0\implies h(z)=C$

$\implies f(x,y,z)=ye^{xz}+C$

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 $C$.