2. CHAPTER 14 Vector Calculus Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14.1VECTOR FIELDS 14.2LINE INTEGRALS 14.3INDEPENDENCE OF PATH AND CONSERVATIVE VECTOR FIELDS 14.4GREEN’S THEOREM 14.5CURL AND DIVERGENCE 14.6SURFACE INTEGRALS 14.7THE DIVERGENCE THEOREM 14.8STOKES’ THEOREM

3. CHAPTER 14 Vector Calculus Slide 3 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 14.9APPLICATIONS OF VECTOR CALCULUS

4. EXAMPLE 14.9APPLICATIONS OF VECTOR CALCULUS 9.1Finding the Flux of a Velocity Field Slide 4 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Suppose that the velocity field v of a fluid has a vector potential w, that is, v = ∇ ×w. Show that v is incompressible and that the flux of v across any closed surface is 0. Also, show that if a closed surface S is partitioned into surfaces S 1 and S 2 (that is, S = S 1 ∪ S 2 and S 1 ∩ S 2 = ∅ ), then the flux of v across S 1 is the additive inverse of the flux of v across S 2.

5. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.1Finding the Flux of a Velocity Field Slide 5 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. To show that v is incompressible, note that ∇ · v = ∇ · ( ∇ ×w) = 0, since the divergence of the curl of any vector field is zero. Next, suppose that the closed surface S is the boundary of the solid Q. Then from the Divergence Theorem,

6. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.1Finding the Flux of a Velocity Field Slide 6 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

7. EXAMPLE 14.9APPLICATIONS OF VECTOR CALCULUS 9.2Computing a Surface Integral Using the Complement of the Surface Slide 7 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the flux of the vector field ∇ ×F across S, where and S is the portion of the cube 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 above the xy-plane.

8. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.2Computing a Surface Integral Using the Complement of the Surface Slide 8 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

9. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.2Computing a Surface Integral Using the Complement of the Surface Slide 9 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

10. 14.9APPLICATIONS OF VECTOR CALCULUS Deriving Fundamental Equations Slide 10 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. One very important use of the Divergence Theorem and Stokes’ Theorem is in deriving certain fundamental equations in physics and engineering. The technique we use here to derive the heat equation is typical of the use of these theorems. In this technique, we start with two different descriptions of the same quantity and use the vector calculus to draw conclusions about the functions involved.

11. 14.9APPLICATIONS OF VECTOR CALCULUS Preliminaries for the Heat Equation Derivation Slide 11 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Net heat flow out of Q: Example 6.7 from physics

12. EXAMPLE 14.9APPLICATIONS OF VECTOR CALCULUS 9.3Deriving the Heat Equation Slide 12 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Use the Divergence Theorem and equation (9.1) to derive the heat equation where α 2 = k/(ρσ) and ∇ 2 T = ∇ · ( ∇ T ) is the Laplacian of T.

13. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.3Deriving the Heat Equation Slide 13 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

14. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.3Deriving the Heat Equation Slide 14 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Observe that the only way for the integral in (9.2) to be zero for every solid Q is for the integrand to be zero.

15. 14.9APPLICATIONS OF VECTOR CALCULUS Preliminaries for the continuity Equation Derivation Slide 15 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Consider a fluid that has density function ρ. We also assume that the fluid has velocity field v and that there are no sources or sinks. The rate of change of mass is:

16. 14.9APPLICATIONS OF VECTOR CALCULUS Preliminaries for the continuity Equation Derivation Slide 16 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. The rate of change of mass can be expressed in another way by considering the flux across ∂Q:

17. EXAMPLE 14.9APPLICATIONS OF VECTOR CALCULUS 9.4Deriving the Continuity Equation Slide 17 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Use the Divergence Theorem and equations (9.3) and (9.4) to derive the continuity equation:

18. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.4Deriving the Continuity Equation Slide 18 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Equate (9.3) and (9.4):

19. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.4Deriving the Continuity Equation Slide 19 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

20. 14.9APPLICATIONS OF VECTOR CALCULUS MAXWELL’S EQUATIONS Slide 20 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

21. EXAMPLE 14.9APPLICATIONS OF VECTOR CALCULUS 9.6Deriving Ampere’s Law Slide 21 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. In the case where E is constant and I represents current, use the relationship to derive Ampere’s law:

22. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.6Deriving Ampere’s Law Slide 22 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Let S be any capping surface for C, that is, any positively oriented two-sided surface bounded by C. The enclosed current I is then related to the current density by

23. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.6Deriving Ampere’s Law Slide 23 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. By Stokes’ Theorem,

24. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.6Deriving Ampere’s Law Slide 24 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25. EXAMPLE 14.9APPLICATIONS OF VECTOR CALCULUS 9.7Using Faraday’s Law to Analyze the Output of a Generator Slide 25 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. An AC generator produces a voltage of 120 sin (120πt) volts. Determine the magnetic flux φ.

26. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.7Using Faraday’s Law to Analyze the Output of a Generator Slide 26 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

27. EXAMPLE Solution 14.9APPLICATIONS OF VECTOR CALCULUS 9.7Using Faraday’s Law to Analyze the Output of a Generator Slide 27 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.