- #1
JD_PM
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- Homework 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):
$$\iint_{R} v \cdot n \frac{dxdz}{|n \cdot j|} = 9\int_{0}^{4} \int_{-2}^{2} (x +x\sqrt{4-x^2})dxdz + 9\int_{0}^{4} \int_{-2}^{2} (-x +x\sqrt{4-x^2})dxdz$$
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
$$v = 9y\hat{i} + 9xy\hat{j} -6z\hat{k}$$
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