Compute divergence theorem for D= 5r2 /4 i in spherical coordinates between r=1 and r=2

To compute the divergence theorem for the given vector field ( textbf{D} = frac{5r^2}{4} hat{i} ) in spherical coordinates between ( r=1 ) and ( r=2 ), we first need to express the vector field in spherical coordinates and then apply the divergence theorem accordingly.

### Step 1: Convert to Spherical Coordinates

The problem presents a vector field in presumably Cartesian coordinates (given the use of ( hat{i} ), typically representing the unit vector in the x-direction in Cartesian coordinates). In spherical coordinates, positions are given by ( (r, theta, phi) ), where:

– (r) is the radial distance from the origin,

– (theta) is the polar angle measured from the z-axis,

– (phi) is the azimuthal angle in the xy-plane from the x-axis.

To convert ( frac{5r^2}{4} hat{i} ) into spherical coordinates, we acknowledge that in spherical coordinates, the Cartesian (x) component relates to (r) as (x = r sin(theta) cos(phi)). However, the vector field provided does not directly correlate with standard spherical components since it’s prescribed in the ( hat{i} ) direction. Therefore, we’re a bit at an impasse regarding conventions; the given description suggests a simplification or a misunderstanding in the application of the vector field