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

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 The Divergence Theorem2

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 The Divergence Theorem3

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 F in this case) over a region to the integral of the original function F over the boundary of the region The Divergence Theorem4

Divergence Theorem Let: ◦ E be a simple solid region and let S be 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 Theorem5

Divergence Theorem Thus, the Divergence Theorem states that: ◦ Under the given conditions, the flux of F across the boundary surface of E is equal to the triple integral of the divergence of F over E The Divergence Theorem6

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 The Divergence Theorem7

History In Eastern Europe, it is known as Ostrogradsky’s Theorem after the Russian mathematician Mikhail Ostrogradsky (1801–1862). ◦ He published this result in The Divergence Theorem8

Example 1 Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S The Divergence Theorem9

Example 2 Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S The Divergence Theorem10

Example 3 Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S The Divergence Theorem11

Example 4 Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S The Divergence Theorem12

Example 5 – pg #11 Use the Divergence Theorem to calculate the surface integral ; that is, calculate the flux of F across S The Divergence Theorem13

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 1 ◦ Example 2 Example The Divergence Theorem14

Demonstrations Feel free to explore these demonstrations below. ◦ The Divergence Theorem The Divergence Theorem ◦ Vector Field with Sources and Sinks Vector Field with Sources and Sinks 16.9 The Divergence Theorem15

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 The Divergence Theorem16

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 boundary of the region The Divergence Theorem17

Fundamental Theorem of Calculus 16.9 The Divergence Theorem18

Fundamental Theorem for Line Integrals 16.9 The Divergence Theorem19

Green’s Theorem 16.9 The Divergence Theorem20

Stokes’ Theorem 16.9 The Divergence Theorem21

Divergence Theorem 16.9 The Divergence Theorem22