Surface Integral (Part 2)

Calculus Level 4

F ⃗ = x ı ^ + y ȷ ^ + z k ^ \Large{\vec{F} = x\hat{\imath} + y\hat{\jmath} + z\hat{k}}

If the surface integral of the vector field F ⃗ \vec{F} over a unit sphere centered on the origin can be expressed as α π \alpha \pi , determine the value of α \alpha .

Note: ı ^ , ȷ ^ , k ^ \hat{\imath}, \hat{\jmath},\hat{k} are unit vectors associated with the three Cartesian coordinate axes ( x , y x,y , and z z respectively).


The answer is 4.

Steven Chase by Steven Chase , 37, USA

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1 solution

First Last
Oct 2, 2016

Because the unit sphere is a nice closed surface, we can use the Divergence Theorem for a quick result.

∬ S F ⃗ d S = ∭ E div F ⃗ d V \displaystyle\iint_S\vec{F}\,d\textbf{S} = \iiint_E\text{div}\vec{F}\, dV

div F ⃗ = div ( x i ^ + y j ^ + z k ^ ) = 3 \displaystyle \text{div}\vec{F} = \text{div}(x\hat{i}+y\hat{j}+z\hat{k}) = 3

∭ E 3 d V = 3 ∭ E d V = 3 4 π 3 = 4 π \displaystyle\iiint_E 3 \, dV = 3\iiint_E dV = 3 \,\frac{4\pi}{3} = \boxed{4\pi}

Where E E is the unit sphere (whose volume is 4 π 3 \frac{4\pi}{3} ).

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