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To apply Stokes’ Theorem to the vector field F = y i + z j + x k, we first need to understand that Stokes’ Theorem relates a surface integral over a surface S to a line integral around the boundary curve C of that surface.
The theorem states:
[
int_C mathbf{F} cdot dmathbf{r} = iint_S (nabla times mathbf{F}) cdot dmathbf{S}
]
1. Calculate the curl of F***:
[
nabla times mathbf{F} = begin{vmatrix}
mathbf{i} & mathbf{j} & mathbf{k} \
frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \
y & z & x
end{vmatrix}
]
Compute the determinant:
[
= mathbf{i}left(frac{partial x}{partial y} – frac{partial z}{partial z}right) – mathbf{j}left(frac{partial x}{partial x} – frac{partial y}{partial z}right) + mathbf{k}left(frac{partial z}{partial x} – frac
Answer: d
Explanation: The curl of y i + z j + x k is i(0-1) – j(1-0) + k(0-1) =
-i –j –k. Since the curl is zero, the value of Stoke’s theorem is zero. The function is said to be irrotational.