To apply the Divergence Theorem, we first need to determine the divergence of the vector field D, which is defined as:

[

mathbf{D} = 2xy , mathbf{i} + x^2 , mathbf{j}

]

The divergence of a vector field (mathbf{D} = P mathbf{i} + Q mathbf{j} + R mathbf{k}) is given by the formula:

[

nabla cdot mathbf{D} = frac{partial P}{partial x} + frac{partial Q}{partial y} + frac{partial R}{partial z}

]

For our field:

– (P = 2xy)

– (Q = x^2)

– (R = 0)

We compute the partial derivatives:

1. (frac{partial P}{partial x} = frac{partial (2xy)}{partial x} = 2y)
2. (frac{partial Q}{partial y} = frac{partial (x^2)}{partial y} = 0)
3. (frac{partial R}{partial z} = frac{partial (0)}{partial z} = 0)

Now putting these together:

[

nabla cdot mathbf{D} = 2y + 0 + 0 =

To compute the divergence theorem for a vector field ( mathbf{D} = frac{5}{4} r^2 mathbf{i} ) in spherical coordinates, we follow these steps:

1. Convert the vector field to spherical coordinates:

In spherical coordinates, the relationship between Cartesian and spherical coordinates is given by:

– ( x = r sin theta cos phi )

– ( y = r sin theta sin phi )

– ( z = r cos theta )

The unit vector ( mathbf{i} ) corresponds to ( sin theta cos phi ), thus:

[

mathbf{D} = frac{5}{4} r^2 hat{r}

]

in spherical coordinates, as ( hat{r} ) is the radial unit vector.

2. Compute the divergence of ( mathbf{D} ):

The divergence in spherical coordinates is given by:

[

nabla cdot mathbf{D} = frac{1}{r^2} frac{partial}{partial r}(r^2 D_r) + frac{1}{r sin theta} frac{partial}{partial theta}(D_theta sin theta) + frac{1}{r} frac{partial D_phi}{

To find the charge enclosed by a sphere of charge density ρ and radius a, you can use the formula for the charge, which is given by:

[ Q = rho cdot V ]

where ( V ) is the volume of the sphere. The volume ( V ) of a sphere is calculated using the formula:

[ V = frac{4}{3} pi r^3 ]

Substituting the radius ( a ) into the volume formula gives:

[ V = frac{4}{3} pi a^3 ]

Therefore, the total charge ( Q ) enclosed within the sphere is:

[ Q = rho cdot frac{4}{3} pi a^3 ]

So, the charge enclosed by the sphere is:

[ Q = frac{4}{3} pi rho a^3 ]