State and explain the divergence theorem.

State and explain the divergence theorem.

asked 2021-03-04
State and explain the divergence theorem.

Answers (1)

The divergence theorem says that if the 3-dimensional space V\subseteq R^{3} and the function F:V\rightarrow R3 satisfy a few nice conditions, then it follows that
\iiintv(\triangledown \times F) dV= \iint∂V (F\times n) dS
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.

Relevant Questions

asked 2021-03-20
The graph of y = f(x) contains the point (0,2), \(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}}={\frac{{-{x}}}{{{y}{e}^{{{x}^{{2}}}}}}}\), and f(x) is greater than 0 for all x, then f(x)=
A) \(\displaystyle{3}+{e}^{{-{x}^{{2}}}}\)
B) \(\displaystyle\sqrt{{{3}}}+{e}^{{-{x}}}\)
C) \(\displaystyle{1}+{e}^{{-{x}}}\)
D) \(\displaystyle\sqrt{{{3}+{e}^{{-{x}^{{2}}}}}}\)
E) \(\displaystyle\sqrt{{{3}+{e}^{{{x}^{{2}}}}}}\)
asked 2021-05-07
Find f'(a)
asked 2021-01-15
Anthony is working for an engineering company that is building a Ferris wheel to be used at county fairs. He wants to create an algebraic model that describes the height of a rider on the wheel in terms of time. He knows that the diameter of the wheel will be 90 feet and that the axle will be built to stand 55 feet off the ground. He also knows they plan to set the wheel to make one rotation every 60 seconds. Write at least two equations that model the height of a rider in terms of t, seconds on the ride, assuming that when t = 0, the rider is at his or her lowest possible height. Explain why both equations are accurate.
Part 2:One of Anthony's co-workers says, "Sine and cosine are basically the same thing." Anthony is not so sure, and can see things either way. Provide one piece of evidence that would confirm the co-worker's point of view. Provide one piece of evidence that would refute it. Hint: It may be helpful to consider the domain and range of different functions, as well as the relationship of each of these functions to triangles in the unit circle
asked 2021-03-09
Find the following limit or state that it does not exist.
asked 2021-02-13
What values(s) of the constant b make \(f(x)=x^{3}−bx\) for \(0\leq x\leq 2\) have:
a.) an absolute min at x=1? Explain.
b.) an absolute max at x=2? Explain.
asked 2021-02-08
Explain the steps you would take to find the inverse of f(x) = 3x − 4. Then find the inverse.
asked 2020-11-10
For \(\displaystyle{f{{\left({x}\right)}}}={\log{
asked 2021-02-27
Form a polynomial f(x) with real coefficients the given degree and zeros. Degree 4, Zeros:, 4-4i, -5 multiplicity 2
asked 2021-03-05
Find the function f given that the slope of the tangent line at any point (x, f(x)) is f '(x) and that the graph of f passes through the given point. \(f '(x)=5(2x − 1)^{4}, (1, 9)\)
asked 2020-11-10
When an electric current passes through two resistors with a resistance r1 and r2. connected in parallel, the combined resistance, R, can be calculated from the equation:
where R, r1 & r2 are greater than 0. Assume that r2 is constant. Show that RR is an increasing function of r1.