What is a constant vector?

A constant vector is one which does not change with time (or any other variable). For example, the origin (0,0,0) is constant, and the point (34,2,22) is constant. They are always in the same place.

A position vector is one that uniquely specifies the position of a point with respect to an origin. For any point P in space, we can define the position vector p = the vector from O to P (where O is the origin). This contrasts with velocity vectors etc. which represent other things.

If A is a constant vector prove that grad(r.A) = A where r is a vector too

Proof

Let r = x i + y j + z k and A = A1 i + A2 j + A3 k


r . A = A1x + A2y + A3z

grad[r . A] = ∂/∂x(A1x) i + ∂/∂y(A2y) j + ∂/∂z(A3z) k

= A1(∂x/∂x) i + A2(∂y/∂y) j + A3(∂z/∂z) k

= A1(1) i + A2(1) j + A3(1) k

grad[r . A] = A1 i + A2 j + A3 k = A

Note; since A is a constant vector ∂/∂x(A1x) = A1(∂x/∂x) and similarly for other two components.

Assignment Question 

Prove that r.A is solenoidal and irrotational 

(Ans: Prove dive (r.A) =0 and Curl (r.A) =0