Chapter 16 – Vector Calculus

1. Chapter 16 – Vector Calculus 16.9 The Divergence Theorem • Objectives: • Understand The Divergence Theorem for simple solid regions. • Use Stokes’ Theorem to evaluate integrals 16.9 The Divergence Theorem

2. Introduction • In Section 16.5, we rewrote Green’s Theorem in a vector version as: where C is the positively oriented boundary curve of the plane region D. 16.9 The Divergence Theorem

3. Equation 1 • If we were seeking to extend this theorem to vector fields on 3, we might make the guess that where S is the boundary surface of the solid region E. 16.9 The Divergence Theorem

4. Introduction • It turns out that Equation 1 is true, under appropriate hypotheses, and is called the Divergence Theorem. • Notice its similarity to Green’s Theorem and Stokes’ Theorem in that: • It relates the integral of a derivative of a function (div Fin this case) over a region to the integral of the original function F over the boundary of the region. 16.9 The Divergence Theorem

5. Divergence Theorem • Let: • E be a simple solid region and let Sbe the boundary surface of E, given with positive (outward) orientation. • F be a vector field whose component functions have continuous partial derivatives on an open region that contains E. • Then, 16.9 The Divergence Theorem

6. Divergence Theorem • Thus, the Divergence Theorem states that: • Under the given conditions, the flux of Facross the boundary surface of E is equal to the triple integral of the divergence ofFoverE. 16.9 The Divergence Theorem

7. History • The Divergence Theorem is sometimes called Gauss’s Theorem after the great German mathematician Karl Friedrich Gauss (1777–1855). • He discovered this theorem during his investigation of electrostatics. 16.9 The Divergence Theorem

8. History • In Eastern Europe, it is known as Ostrogradsky’s Theorem after the Russian mathematician Mikhail Ostrogradsky (1801–1862). • He published this result in 1826. 16.9 The Divergence Theorem

9. Example 1 • Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S. 16.9 The Divergence Theorem

10. Example 2 • Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S. 16.9 The Divergence Theorem

11. Example 3 • Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S. 16.9 The Divergence Theorem

12. Example 4 • Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S. 16.9 The Divergence Theorem

13. Example 5 – pg. 1157 #11 • Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S. 16.9 The Divergence Theorem

14. More Examples The video examples below are from section 16.9 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. • Example 1 • Example 2 16.9 The Divergence Theorem

15. Demonstrations • Feel free to explore these demonstrations below. • The Divergence Theorem • Vector Field with Sources and Sinks 16.9 The Divergence Theorem

16. Review of Chapter • The main results of this chapter are all higher-dimensional versions of the Fundamental Theorem of Calculus (FTC). • To help you remember them, we collect them here (without hypotheses) so that you can see more easily their essential similarity. 16.9 The Divergence Theorem

17. Review of Chapter • In each case, notice that: • On the left side, we have an integral of a “derivative” over a region. • The right side involves the values of the original function only on the boundaryof the region. 16.9 The Divergence Theorem

18. Fundamental Theorem of Calculus 16.9 The Divergence Theorem

19. Fundamental Theorem for Line Integrals 16.9 The Divergence Theorem

20. Green’s Theorem 16.9 The Divergence Theorem

21. Stokes’ Theorem 16.9 The Divergence Theorem

22. Divergence Theorem 16.9 The Divergence Theorem