Let $\varphi$ be a scalar function. Then $\nabla \varphi$ is

The value of the surface integral

where S is the external surface of the sphere x² + y² + z² = R² is

A vector field

𝐁(𝑥, 𝑦, 𝑧) = 𝑥 𝑖̂ + 𝑦 ĵ − 2𝑧 k̂

is defined over a conical region having height ℎ = 2, base radius 𝑟 = 3 and axis along z, as shown in the figure. The base of the cone lies in the x-y plane and is centered at the origin.

If 𝒏 denotes the unit outward normal to the curved surface 𝑆 of the cone, the value of the integral

$\rm \int_SB.n\ dS$

equals _________ . (Answer in integer)