Given the vector A = (cos x)(sin y)âx + (sin x)(cos y) ây, ây denote unit vectors along x,y directions, respectively. The magnitude of the curl of 𝑨 is ______.
Concept:
Let \(\vec A = {A_x}\;{a_x} + {A_y}{a_y} + {A_z}\;{a_z}\)
The curl of A is evaluated as:
\(Curl\;\vec A = \left| {\begin{array}{*{20}{c}} {{a_x}}&{{a_y}}&{{a_z}}\\ {\frac{\partial }{{{\partial x}}}}&{\frac{\partial }{{{\partial y}}}}&{\frac{\partial }{{{\partial z}}}}\\ {{A_x}}&{{A_y}}&{{A_z}} \end{array}} \right|\)
Analysis:
Given
\(\vec A\) = (cos x) (sin y) ax + (sin x) (cos y) ây + 0⋅az
Ax = cos x sin y
Ay = sin x cos y
Az = 0
\(\nabla \times \vec A = \left| {\begin{array}{*{20}{c}} {{a_x}}&{{a_y}}&{{a_z}}\\ {\frac{\partial }{{{\partial x}}}}&{\frac{\partial }{{{\partial y}}}}&{\frac{\partial }{{{\partial z}}}}\\ {\cos x\sin y}&{\sin x\cos y}&0 \end{array}} \right|\)
\(\nabla \times \vec A = {a_x}\left( {0 - 0} \right) - {a_y}\left( {0 - 0} \right) + {a_z}\left( {\cos x\cos y - \cos x\cos y} \right)\)
\(\nabla \times \vec A = 0\;{a_x} - 0\;{a_y} - 0\;{a_z}\)
\(\left| {\nabla \times \vec A} \right| = 0\)
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