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To apply Stokes’ Theorem for the vector field (mathbf{A} = x mathbf{i} + y mathbf{j} + z mathbf{k}), we need to compute the curl of (mathbf{A}) and the line integral of (mathbf{A}) around a closed curve that bounds a surface.
1. Calculate the curl (nabla times mathbf{A}):
[
nabla times mathbf{A} = begin{vmatrix}
mathbf{i} & mathbf{j} & mathbf{k} \
frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \
x & y & z
end{vmatrix}
]
Calculating this determinant:
[
nabla times mathbf{A} = mathbf{i}left(frac{partial z}{partial y} – frac{partial y}{partial z}right) – mathbf{j}left(frac{partial z}{partial x} – frac{partial x}{partial z}right) + mathbf{k}left(frac{partial y}{partial x} – frac{partial x}{partial y}right)
]
Evaluating the partial derivatives, we get:
[
nabla times mathbf{A} = 0
Answer: d
Explanation: Since curl is required, we need not bother about divergence property. The curl of the function will be i(0-0) – j(0-0) + k(0-0) = 0. The curl is zero, thus the function is said to be irrotational or curl free.