State and explain the divergence theorem.

Question

yunitsiL · Accepted Answer

The divergence theorem says that if the 3-dimensional space $V \subseteq R^{3}$ and the function F: $V \to R^{3}$ satisfy a few nice conditions, then it follows that
$\int \int \int_{V} (▽ \cdot F) d V = \int \int_{\partial V} (F \cdot n) d S$
where ∂V is the boundary of the space V. I'd recommend looking at your textbook to see exactly how the conditions on V and F are stated in your course.
To understand the meaning of the formula, we have to understand what each integral is measuring. On the LHS, we have the integral of the divergence of F throughout V. Roughly speaking, the divergence of a vector field measures how much the vector field "spreads out" at a given point. So the integral, being an aggregate of this information, gives an idea of the overall tendency of the vector field to "spread out" in the space V.
On the RHS, we have the flux integral over the boundary of V. This integral measures how much F "points out of" the space V by totaling up the contributions everywhere along the boundary.
So the fact that these are equal is pretty cool in that it says if you want to know this general tendency of the vector field F everywhere throughout V, then it suffices to just see how much it points out of vs into the space only along the boundary.

State and explain the divergence theorem.

Answered question

Answer & Explanation

New Questions in Algebra I