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

`\\abla f(x,y,z)=\mathbf f(x,y,z)=yze^(xz)\,\mathbf i+e^(xz)\,\mathbf j+xye^(xz)\,\mathbf k`


`(\partial f(x,y,z))/(\partial x)=yze^(xz)`

`\implies f(x,y,z)=\displaystyle\int yze^(xz)\,\mathrm dx=\frac{yz}ze^(xz)+g(y,z)`

`f(x,y,z)=ye^(xz)+g(y,z)`


`(\partial f(x,y,z))/(\partial y)=e^(xz)=e^(xz)(\partial g(y,z))/(\partial y)`

`\implies(\partial g(y,z))/(\partial y)=0\implies g(y,z)=h(z)`

`f(x,y,z)=ye^(xz)+h(z)`


`(\partial f(x,y,z))/(\partial z)=xye^(xz)=xye^(xz)+(\partial h(z))/(\partial z)`

`\implies(\partial h(z))/(\partial z)=0\implies h(z)=C`

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