Question

F(x, y, z) = yz i + xz j + (xy + 18z) k c is the line segment from (1, 0, −1) to (5, 6, 1) (a) find a function f such that f = ∇f. f(x, y, z) = (b) use part (a) to evaluate c ∇f · dr along the given curve
c.

Answer 1

$\nabla f(x,y,z)=\dfrac{\partial f}{\partial x}\,\mathbf i+\dfrac{\partial f}{\partial y}\,\mathbf j+\dfrac{\partial f}{\partial z}\,\mathbf k=\mathbf f(x,y,z)$

$\displaystyle\int\frac{\partial f}{\partial x}\,\mathrm dx=\int yz\,\mathrm dx$
$\implies f(x,y,z)=xyz+g(y,z)$
$\dfrac{\partial f}{\partial y}=xz+\dfrac{\partial g}{\partial z}=xz$
$\implies\dfrac{\partial g}{\partial z}=0$
$\implies g(y,z)=h(z)$

$\dfrac{\partial f}{\partial z}=xy+\dfrac{\mathrm dh}{\mathrm dz}=xy+18z$
$\implies\dfrac{\mathrm dh}{\mathrm dz}=18z$
$\implies h(z)=9z^2+C$

$\implies f(x,y,z)=xyz+9z^2+C$

$\implies\displaystyle\int_{\mathcal C}\mathbf f\cdot\mathrm d\mathbf r=f(5,6,1)-f(1,0,-1)=30$