# Important Conclusions on the Basis of Gradient, Divergence and Curl

• 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

• 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.