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

To compute the divergence of a vector field using the divergence theorem in spherical coordinates, we follow a systematic approach. Given a vector field (mathbf{D} = frac{5r^2}{4} hat{i}) in a coordinate system, we first need to express this field in spherical coordinates and then find its divergence. However, a direct conversion of the given vector field into spherical coordinates poses a challenge since the field is given in a form that suggests it’s already partially in a non-Cartesian form ((frac{5r^2}{4} hat{i}) suggests a dependence on radial distance but uses (hat{i}), which is a Cartesian unit vector). Assuming the intention is to deal with a radially dependent vector field in a spherical context, we can reinterpret the vector field in spherical coordinates, focusing on its radial component only.

Spherical Coordinates Background

In spherical coordinates, a position in space is given by (r) (radial distance), (theta) (polar angle, measured from the positive z-axis), and (phi) (azimuthal angle, measured in the x-y plane from the positive x-axis). Vector fields in spherical coordinates are expressed in terms of these variables and their unit vectors (hat{r}), (hat{theta}), and (hat{phi}).

Given Vector Field

Given the ambiguities in the initial presentation of the vector field, but