• The gradient of scalar field θ, i.e., ∆θ is a vector.
• The divergence of a vector field A , i .e., V.A is a scalar.
• The curl of a vector field A, i.e.,∆x A is a vector.
• The divergence of the gradient of a scalar field θ may be given as
where ∆2 is known as the Laplacian operator.
• The curl of the gradient of a scalar field is zero, i.e.;
curl (grad θ) = ∆ x (∆ θ) = 0.
It is because the cross-product of.two similar vectors(\?) is zero.
• The divergence of curl of a vector field is zero, i.e.,
div( curl A)= ∆.(∆ x A)= 0
It is because the vector ∆ x A is perpendicular to VI and therefore the dot product of
V7 with a vector perpendicular to it will be zero.
• The curl of the curl of a vector field is defined as
curl( curl A) = ∆ x(∆ x A) = ∆ (∆.A)- ∆ 2A
=grad (div A)- ∆ A
• Solenoidal Vector Point Function
When the divergence of a vector point function A is zero, i.e.,
div =∆ .A= 0
then it is said to be solenoidal vector point function. Thus the vector point function will be solenoidal over a region, if the flux across any closed surface in that region is zero.
• Irrotational Vector Point Function
If the curl of a vector point function is zero it is said to be irrotational i.e., when
curl ∆=∆ xA=O
then A is irrotational.