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To find the curl of the vector field ( mathbf{A} = yz hat{i} + 4xy hat{j} + y hat{k} ), we use the formula for curl in three dimensions:
[
nabla times mathbf{A} = begin{vmatrix}
hat{i} & hat{j} & hat{k} \
frac{partial}{partial x} & frac{partial}{partial y} & frac{partial}{partial z} \
yz & 4xy & y
end{vmatrix}
]
Calculating this determinant, we have:
[
nabla times mathbf{A} = hat{i} left( frac{partial y}{partial y} – frac{partial}{partial z}(4xy) right) – hat{j} left( frac{partial y}{partial x} – frac{partial}{partial z}(yz) right) + hat{k} left( frac{partial (4xy)}{partial x} – frac{partial (yz)}{partial y} right)
]
Calculating each term:
1. For the ( hat{i} ) component:
[
frac{partial y}{partial y} = 1, quad frac{partial (4xy)}{
Answer: d
Explanation: Curl A = i(Dy(y) – Dz(0)) – j (Dx(0) – Dz(yz)) + k(Dx(4xy) – Dy(yz)) =
i + y j + (4y – z)k.