 Solution

ID: past-exam-of-the-mathematical-methods-ii-course-of-the-university-of-oxford/2014-resit/11/c/solution

Take the curl component by component:

\nabla \times F = (\frac{\partial ( 3 x + cy - 2 z )}{\partial y} - \frac{\partial ( b x + y - z )}{\partial z}, \frac{\partial ( x + 3 y + a z )}{\partial z} - \frac{\partial ( 3 x + cy - 2 z )}{\partial x}, \frac{\partial ( b x + y - z )}{\partial x} - \frac{\partial ( x + 3 y + a z )}{\partial y}) = (c + 1, a - 3, b - 3) .

The field is irrotational exactly when all three constant components vanish:

a = 3, b = 3, c = - 1.

 New to topics? Read the docs here!