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To find the curl of the vector field ( mathbf{A} = (y cos(ax)) mathbf{i} + (y + e^x) mathbf{k} ), we can use the curl formula in three dimensions:
[
text{curl} , mathbf{A} = nabla times mathbf{A} = left( frac{partial A_z}{partial y} – frac{partial A_y}{partial z} right) mathbf{i} – left( frac{partial A_z}{partial x} – frac{partial A_x}{partial z} right) mathbf{j} + left( frac{partial A_y}{partial x} – frac{partial A_x}{partial y} right) mathbf{k}
]
Given:
– ( A_x = y cos(ax) )
– ( A_y = 0 )
– ( A_z = y + e^x )
Now, compute each component:
1. First Component (i):
[
frac{partial A_z}{partial y} = frac{partial}{partial y}(y + e^x) = 1
]
[
frac{partial A_y}{partial z} = frac{partial}{partial z}(0) = 0
Answer: b
Explanation: Curl A = i(Dy(y + ex)) – j (Dx(y + ex) – Dz(y cos ax)) + k(-Dy(y cos ax))
= 1.i – j(ex) – k cos ax = i – ex j – cos ax k.