Given a function φ = 1/2 (x² + y² + z²) in three-dimensional Cartesian space, the value of the surface integral ∯_S n̂ · ∇φ dS, where S is the surface of a sphere of unit radius and n̂ is the outward unit normal vector on S, is

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Accepted Answer

4π. The surface integral evaluates the flux of the gradient of the function φ through the surface of the sphere. Since the gradient ∇φ points radially outward and has a constant magnitude on the surface of the unit sphere, the integral simplifies to the product of the magnitude of the gradient at the surface and the surface area of the sphere, resulting in a value of 4π.

Given a function φ = 1/2 (x² + y² + z²) in three-dimensional Cartesian space, the value of the surface integral ∯_S n̂ · ∇φ dS, where S is the surface of a sphere of unit radius and n̂ is the outward unit normal vector on S, is

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