Calculus III

Vector operators
Applications
Curl and divergence of a vector

In Physics, being the velocity vector field of a moving object, this object is irrotational when curl () = and incompressible when div () = 0.

1. Curl of a vector

Given the vector field = P + Q + R , the curl of the vector is defined by the cross product:

Curl () = ∇ ×

∇ = ∂ is the partial differential operator. That is the gradient vector, that acts on a scalar as a function, to give a vector.

∇ = ∂ = (∂/∂x) + (∂/∂y) + (∂/∂z)

Acting on a function f, yields:

∇f = (∂f/∂x) + (∂f/∂y) + ((∂f/∂z)

Therefore

The curl of a vector is the cross product of the gradient vector ∇ and the vector itself.

∇ × =
		∂x	∂y	∂z
		P	Q	R

= (R_y - Q_x) + (P_z - R_x) + (Q_x - P_y)

2. Properties of the curl operator

• If f(x,y,z) has continuous second order partial derivatives then curl(∇f) = .

• If is a conservative vector field then curl() = . That is the definition of a conservative vector field.

• If is defined on , has its components continuous first order partial derivative, and curl () = , then is a conservative vector field.

3. Divergence operator

Given the vector field = P + Q + R , the divergence of the vector is defined by the dot product:

div () = ∇ .

∇ = ∂ is the partial differential operator. That is the gradient vector, that acts on a vector to give a scalar.

∇ = ∂ = (∂/∂x) + (∂/∂y) + (∂/∂z)

Acting on a vector , yields:

∇ . , = ∂P/∂x + ∂Q/∂y + ∂R/∂z

Therefore

The divergence of a vector is the dot product of the gradient vector ∇ and the vector itself.

4. Property of divergence-curl operator

• Relationship between the curl and the divergence:

div (curl ) = 0