The surface integral $$ \iint_S \mathbf{F} \cdot d\mathbf{S} $$ where $ \mathbf{F} = x\hat{i} + y\hat{j} - z\hat{k} $ and $ S $ is the surface of the cylinder $ x^2 + y^2 = 4 $ bounded by the planes $ z = 0 $ and $ z = 4 $, equals:

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We calculate the surface integral using the given vector field and the surface of the cylinder. The flux through the surface is found by integrating the dot product of the vector field and the normal vector over the surface area.

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$ \frac{32}{3} $

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	LIST I (Type of the Matrix)	LIST II (Property)
A.	Symmetric Matrix	I. a_ij = a_ji, for values of i and j
B.	Hermitian Matrix	II. a_ij = ā_ji, for values of i and j
C.	Skew-Hermitian matrix	III. a_ij = -ā_ji, for values of i and j
D.	Skew-Symmetric matrix	IV. a_ij = -a_ji, for values of i and j

The surface integral \[ \iint_S \mathbf{F} \cdot d\mathbf{S} \] where \( \mathbf{F} = x\hat{i} + y\hat{j} - z\hat{k} \) and \( S \) is the surface of the cylinder \( x^2 + y^2 = 4 \) bounded by the planes \( z = 0 \) and \( z = 4 \), equals:

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The Correct Option is C

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