Question

F(x, y, z) = yzexzi + exzj + xyexzk,c: r(t) = (t2 + 4)i + (t2 − 1)j + (t2 − 2t)k, 0 ≤ t ≤ 2(a) find a function f such that f = ∇f.

Answer 1

$\nabla f(x,y,z)=\mathbf f(x,y,z)=yze^{xz}\,\mathbf i+e^{xz}\,\mathbf j+xye^{xz}\,\mathbf k$

$\dfrac{\partial f(x,y,z)}{\partial x}=yze^{xz}$
$\implies f(x,y,z)=\displaystyle\int yze^{xz}\,\mathrm dx=\dfrac{yz}ze^{xz}+g(y,z)$
$f(x,y,z)=ye^{xz}+g(y,z)$

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

$\dfrac{\partial f(x,y,z)}{\partial z}=xye^{xz}=xye^{xz}+\dfrac{\partial h(z)}{\partial z}$
$\implies\dfrac{\partial h(z)}{\partial z}=0\implies h(z)=C$
$f(x,y,z)=ye^{xz}+C$