CHAPTER 14 Vector Calculus Slide 2 © The McGraw-Hill Companies, Inc. Permission required for...
-
Upload
melina-golden -
Category
Documents
-
view
219 -
download
0
description
Transcript of CHAPTER 14 Vector Calculus Slide 2 © The McGraw-Hill Companies, Inc. Permission required for...
CHAPTER
14Vector Calculus
14
Slide 2© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
14.1 VECTOR FIELDS14.2 LINE INTEGRALS14.3 INDEPENDENCE OF PATH AND CONSERVATIVE
VECTOR FIELDS14.4 GREEN’S THEOREM14.5 CURL AND DIVERGENCE14.6 SURFACE INTEGRALS14.7 THE DIVERGENCE THEOREM14.8 STOKES’ THEOREM
CHAPTER
14Vector Calculus
14
Slide 3© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
14.9 APPLICATIONS OF VECTOR CALCULUS
EXAMPLE
14.9 APPLICATIONS OF VECTOR CALCULUS
9.1 Finding 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 S1 and S2 (that is, S = S1 ∪ S2 and S1 ∩ S2 = ), ∅then the flux of v across S1 is the additive inverse of the flux of v across S2.
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.1 Finding 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,
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.1 Finding the Flux of a Velocity Field
Slide 6© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
14.9 APPLICATIONS OF VECTOR CALCULUS
9.2 Computing 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.
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.2 Computing a Surface Integral Using the Complement of the Surface
Slide 8© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.2 Computing a Surface Integral Using the Complement of the Surface
Slide 9© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
14.9 APPLICATIONS 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.
14.9 APPLICATIONS 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
EXAMPLE
14.9 APPLICATIONS OF VECTOR CALCULUS
9.3 Deriving 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 α2 = k/(ρσ) and ∇2T = · (∇ ∇T ) is the Laplacian of T.
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.3 Deriving the Heat Equation
Slide 13© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.3 Deriving 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.
14.9 APPLICATIONS 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:
14.9 APPLICATIONS 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:
EXAMPLE
14.9 APPLICATIONS OF VECTOR CALCULUS
9.4 Deriving 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:
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.4 Deriving the Continuity Equation
Slide 18© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Equate (9.3) and (9.4):
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.4 Deriving the Continuity Equation
Slide 19© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
14.9 APPLICATIONS OF VECTOR CALCULUS
MAXWELL’S EQUATIONS
Slide 20© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
14.9 APPLICATIONS OF VECTOR CALCULUS
9.6 Deriving 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:
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.6 Deriving 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-sidedsurface bounded by C.
The enclosed current I is then related to the current density by
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.6 Deriving Ampere’s Law
Slide 23© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
By Stokes’ Theorem,
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.6 Deriving Ampere’s Law
Slide 24© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
14.9 APPLICATIONS OF VECTOR CALCULUS
9.7 Using 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.
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.7 Using Faraday’s Law to Analyze the Output of a Generator
Slide 26© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
EXAMPLE
Solution
14.9 APPLICATIONS OF VECTOR CALCULUS
9.7 Using Faraday’s Law to Analyze the Output of a Generator
Slide 27© The McGraw-Hill Companies, Inc. Permission required for reproduction or display.