Vector Fields and Divergence

9 Questions

Is the divergence of the vector field P equal to 2XYZ + X?

True

What is the divergence of the vector field P in cartesian coordinates?

2XYZ + X

What is the divergence of the vector field P in cylindrical coordinates?

2XYZ

What does Stoke's theorem state?

The circulation of a vector field A around a closed path I is equal to the surface integral of the curl of A over the open surface bounded by L.

Is Stoke's theorem applicable only if A and △XA are continuous on S?

Yes

What is the relationship between the circulation of a vector field A and the surface integral of the curl of A?

They are equal

What is Stoke's theorem?

The circulation of a vector field $A$ around a closed path $I$ is equal to the surface integral of the curl of $A$ over the open surface bounded by $L$, provided $A$ and $\Delta XA$ are continuous on $S$.

Applicability of Stoke's theorem

Yes, Stoke's theorem is applicable only if $A$ and $\Delta XA$ are continuous on $S$.

Relationship between circulation and surface integral

The circulation of a vector field $A$ is equal to the surface integral of the curl of $A$.

Study Notes

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.