Vector Fields and Divergence

Divergence of a Vector Field

• The divergence of the vector field P is not equal to 2XYZ + X, as the expression lacks a differential operator.

Divergence in Cartesian Coordinates

• The divergence of the vector field P in Cartesian coordinates is given by ∇⋅P = (∂P_x/∂x) + (∂P_y/∂y) + (∂P_z/∂z).

Divergence in Cylindrical Coordinates

• The divergence of the vector field P in cylindrical coordinates is given by ∇⋅P = (1/r) (∂(rP_r)/∂r) + (1/r) (∂P_θ/∂θ) + (∂P_z/∂z).

Stoke's Theorem

• Stoke's theorem relates the circulation of a vector field around a closed curve to the surface integral of the curl of that vector field.

Applicability of Stoke's Theorem

• Stoke's theorem is applicable only if the vector field A and its first partial derivatives are continuous on the surface S and its boundary.

Relationship between Circulation and Surface Integral

• The circulation of a vector field A around a closed curve is equal to the surface integral of the curl of A, i.e., ∯A⋅dl = ∬ (∇×A) ⋅dS.

This quiz tests your knowledge of vector fields and divergence in Cartesian coordinates. The question asks whether the divergence of a given vector field is equal to a specific expression.