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To determine if the vector field E = yz i + xz j + xy k is irrotational, we need to compute the curl of the vector field. A vector field is irrotational if its curl is zero.
The curl of a vector field F = P i + Q j + R k is given by:
∇ × F = (∂R/∂y – ∂Q/∂z) i + (∂P/∂z – ∂R/∂x) j + (∂Q/∂x – ∂P/∂y) k
For our vector field E:
– P = yz
– Q = xz
– R = xy
Now we compute the partial derivatives:
1. ∂R/∂y = ∂(xy)/∂y = x
2. ∂Q/∂z = ∂(xz)/∂z = x
3. ∂P/∂z = ∂(yz)/∂z = y
4. ∂R/∂x = ∂(xy)/∂x = y
5. ∂Q/∂x = ∂(xz)/∂x = z
6. ∂P/∂y = ∂(yz)/∂y = z
Now substituting these into the curl formula:
Answer: a
Explanation: Curl E = i(Dy(xy) – Dz(xz)) – j (Dx(xy) – Dz(yz)) + k(Dx(xz) – Dy(yz)) =
i(x – x) – j(y – y) + k(z – z) = 0
Since the curl is zero, the vector is irrotational or curl-free.