The vector function F(x,y,z)= (e^2yz) i+(e^3xz) j +( e^xy ) k How would you calculate curl F at (1,1,1)?

Let F(x, y, z) = F1 i + F2 j + F3 k, then

Here, F(x, y, z) = (e^2yz) i+(e^3xz) j +( e^xy ) k

So, F1 = e^2yz, F2 = e^3xz, F3 = e^xy

By definition,

curl(F) = (d/dx i + d/dy j + d/dz k)*(F1 i + F2 j + F3 k)

[cross product of two vectors]

curl(F)

= d/dx (F2) k - d/dx (F3) j - d/dy (F1) k + d/dy (F3) i + d/dz (F1) j - d/dz (F2) i

= {d/dy (F3) - d/dz (F2)} i + {d/dz (F1) - d/dx (F3)} j + {d/dx (F2) - d/dy (F1)} k

[Note that 'd' stands for 'del'.]

Substituting the expressions for F1, F2 & F3, curl(F)

= {d/dy (e^xy) - d/dz (e^3xz)} i + {d/dz (e^2yz) - d/dx (e^xy)} j + {d/dx (e^3xz) - d/dy (e^2yz)} k

Point (1,1,1) refers to: x = y = z = 1

So, curl(F) at (1, 1, 1)

= {(e) - (3e^3)} i + {(2e^2) - (e)} j + {(3e^3) - (2e^2)} k

Hence (f) is correct.

there's a vector operator talked approximately as "nabla" the logo is an inverted triangle(I´ll useD) D= d/dx i+d/dy j+d/dz ok if your vector is V= Pi+Qj+Rk ,curl V is the vector manufactured from D and V