Surface integral - undefined vector field

Question

Consider the vector field $$\mathbf F(x,y, z) = \frac{x\hat{i} + y\hat{j} + z\hat{k}}{[x^2+y^2+z^2]^{3/2}}$$

Let $S_1$ be the sphere given by $x^2 + (y-2)^2 + z^2 = 9$ oriented outwards. Compute $$\iint_{S_1}\mathbf{F}\mathbf{\cdot} \hat{\mathbf n}\ dS$$

I understand that since $\mathbf{F}$ is not defined at the origin, we cannot directly apply the divergence theorem to this. However, is there still a simpler way of doing this than actually computing the integral as a whole?

It might be relevant that $(x\hat i + y\hat j + z \hat k)/\sqrt{x^2 + y^2 + z^2}$ is the unit vector pointing outwards from the origin at all points. — Andrew Li, Apr 25 '19 at 22:08
Was my answer helpful to you, or do you need some further clarification? — Spencer, Apr 30 '19 at 06:03

score 0 · Answer 1 · answered Apr 25 '19 at 23:44

The vector field is one that is common in electrostatics. It is the electric field for a point charge which is located at the origin. The divergence of this field is 0 everywhere except where it is undefined (at the origin).

The source of the flux is only the singularity at the origin. Any other surface will have the same flux as long as it contains that singularity.

$$ \iint_{S_1} \vec{F} \cdot \hat{n} dA = \iint_{S_2} \vec{F} \cdot \hat{n} dA $$ $$\text{(For any two closed surfaces which enclose the origin)}$$

The easiest surface to use would be a unit sphere centered on the origin.

For more information on this look up Coulomb's law and Gauss' law for electrostatic fields.

Surface integral - undefined vector field

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