Question

In cylindrical Coordinates, does the below statement hold true or false, following the preamble below the statement?

∇f=∂f/∂u i + ∂f/∂v j + ∂f/∂w k=a(u,v,w)u+b(u,v,w)v+c(u,v,w)w

where

u,v, and w

are orthonormal

basis vectors in the new coordinate system and

a,b,andc

are continuous functions.

Answer 1

In cylindrical Coordinate system, the above statement doesn't hold good.

let i, j, k are unit vectors in cartesian system. u, v, w are the unit vectors in cylindrical coordinate system.

Let T is a scalar function.

The gradient in the cartesian system can be written as

(1)

For obtaining the gradient is cylindrical coordinate system, let us first write the unit vectors of cartesian coordinate system in terms of the cylindrical coordinate system.

(2)

(3)

The third vector is same for both the coordinate system

Using the coordinate relationship,

(4)

(5)

Using the differential calculus, we can write

(6)

(7)

Thus using the above relations and replacing the above in (1) gives

(8)