The derivative of a vector P according to a scalar variable t is: 

The derivative of the sum of two vectors is: 

The derivative of the product of a vector P and a scalar u(t)according to t is: 

The derivative of two vectors dot product: 

For the cross product the derivative is: 

Gradient, If ϕ is a scalar function defined by ϕ=f(x,y,z), we define the gradient of
ϕ, that is a vector in the ndirection and represents the maximum space rate of change of ϕ.

The del operator: 

Gradient operator: 




divergence, when the vector operator ᐁ is dotted into a vector V, the result is the divergence of V. 
Note that:

Curl, When the vector operator ᐁ is crossed into a vector V, the result is the curl of V. 
Curl, When the vector operator ᐁ is crossed into a vector V, the result is the curl of V.

Laplacian operator is the dot product of the vector ᐁ into itself gives the scalar operator known as Laplacian operator. 

Other relations of the ᐁ operator. 
ᐁ ✕ (ᐁΦ ) = 0 
ᐁ· (ᐁ✕A ) = 0 
ᐁ✕(ᐁ✕A ) = ᐁ(ᐁ·A)−ᐁ^{2}A 
ᐁ(Φ + ψ ) = ᐁΦ + ᐁψ 
ᐁ·(A + B ) = ᐁ·A + ᐁ·B 
ᐁ✕(A + B ) = ᐁ✕A + ᐁ✕B 
ᐁ·(ΦA ) = (ᐁΦ)·A + Φ(ᐁ·A ) 
ᐁ✕(ΦA) = (ᐁΦ)✕A + Φ(ᐁ✕A ) 
ᐁ· (A✕B ) = B · (ᐁ✕A ) − A·(ᐁ✕B ) 
ᐁ✕(A✕B ) = (B ·ᐁ)A − B(ᐁ·A ) − (A·ᐁ)B + A(ᐁ·B ) 
ᐁ(A·B ) = (B·ᐁ)A + (A·ᐁ)B + B✕(ᐁ✕A ) A✕(ᐁ✕B ) 
