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- Problem Statement
- ##v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}##

- Relevant Equations
- ##\int_{V}\nabla \cdot v dV = \int_{S} v \cdot da##

I am checking the divergence theorem for the vector field:

$$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$

The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2##

This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region (volume); using cylindrical coordinates:

$$\int_{0}^{4}\int_{0}^{2\pi}\int_{0}^{2} (9rcos\theta - 6)r dr d \theta dz = -96\pi$$

Now the right hand side of the divergence theorem; the value of the function at the boundary(surface; i.e. its flux):

$$\iint_{R} v \cdot n \frac{dxdz}{|n \cdot j|} = 9\int_{0}^{4} \int_{-2}^{2} (x +x\sqrt{4-x^2})dxdz + 9\int_{0}^{4} \int_{-2}^{2} (-x +x\sqrt{4-x^2})dxdz$$

Thanks

$$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$

The region is inside the cylinder ##x^2 + y^2 = 4## and between ##z = 0## and ##z = x^2 + y^2##

This is my set up for the integral of the derivative (##\nabla \cdot v##) over the region (volume); using cylindrical coordinates:

$$\int_{0}^{4}\int_{0}^{2\pi}\int_{0}^{2} (9rcos\theta - 6)r dr d \theta dz = -96\pi$$

Now the right hand side of the divergence theorem; the value of the function at the boundary(surface; i.e. its flux):

$$\iint_{R} v \cdot n \frac{dxdz}{|n \cdot j|} = 9\int_{0}^{4} \int_{-2}^{2} (x +x\sqrt{4-x^2})dxdz + 9\int_{0}^{4} \int_{-2}^{2} (-x +x\sqrt{4-x^2})dxdz$$

**Is this later integral well arranged?**##9\int_{0}^{4} \int_{-2}^{2} (x +x\sqrt{4-x^2})dxdz##**is vanishing and I don't get**##-96\pi##**in the right hand side.**Thanks