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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Surface integrals can be simplified by using symmetry and parameterizing the surface in cylindrical coordinates
The surface integral is determined using the provided vector field and the cylinder's surface. Flux through the surface is calculated by integrating the dot product of the vector field and the normal vector over the surface area.\r