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


14.9 APPLICATIONS OF VECTOR CALCULUS

EXAMPLE


9.1 Finding the Flux of a Velocity Field


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




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




EXAMPLE


9.2 Computing a Surface Integral Using the Complement of the Surface


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





Deriving Fundamental Equations


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.


Preliminaries for the Heat Equation Derivation


Net heat flow out of Q:

Example 6.7

from physics

EXAMPLE


9.3 Deriving the Heat Equation


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




EXAMPLE

Solution




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.


Preliminaries for the continuity Equation Derivation


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:


Preliminaries for the continuity Equation Derivation


The rate of change of mass can be expressed in another way by considering the flux across ∂Q:

EXAMPLE


9.4 Deriving the Continuity Equation


Use the Divergence Theorem and equations (9.3) and (9.4) to derive the continuity equation:

EXAMPLE

Solution




Equate (9.3) and (9.4):

EXAMPLE

Solution





MAXWELL’S EQUATIONS


EXAMPLE


9.6 Deriving Ampere’s Law


In the case where E is constant and I represents current, use the relationship

to derive Ampere’s law:

EXAMPLE

Solution




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




By Stokes’ Theorem,

EXAMPLE

Solution




EXAMPLE


9.7 Using Faraday’s Law to Analyze the Output of a Generator


An AC generator produces a voltage of 120 sin (120πt) volts.

Determine the magnetic flux φ.

EXAMPLE

Solution




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