The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = x i + y j + z k defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors.
∬_S (F · n) dA
where S is the sphere, x² + y² + z² = 1 and n is the o

Question

The following surface integral is to be evaluated over a sphere for the given steady velocity vector field F = x i + y j + z k defined with respect to a Cartesian coordinate system having i, j and k as unit base vectors.
∬_S (F · n) dA
where S is the sphere, x² + y² + z² = 1 and n is the outward unit normal vector to the sphere. The value of the surface integral is

Accepted Answer

4. The surface integral evaluates the flux of the vector field F through the sphere's surface. Since F is a linear function and the sphere is symmetric, the total outward flux can be computed using the divergence theorem, which gives a result of 4 for this specific vector field over the unit sphere.

Solution

Related GATE Engineering Mathematics questions