Vector FieldsMathematics 55 - Elementary Analysis 3

Institute of MathematicsUniversity of the Philippines-Diliman

1 / 20

Introduction

The following shows air velocity vectors that indicate the wind speed anddirection at points above the surface elevation in Philippine area onMarch 16, 2009 00 :00 UTC.

To each point in the air we associate a vector (wind velocity). This is an

example of a vector field

2 / 20

Vector Fields

Definition

A vector field on R2 (or R3) is a function~F that assigns to each (x,y)(or (x,y,z)) a two (or three) dimensional vector~F(x,y) (or~F(x,y,z)).Representations of the vector~F(x,y) (or~F(x,y,z)) withinitial point (x,y) (or (x,y,z)) are called flowlines of the vector field.

Notations :

~F(x,y)= P(x,y)i+Q(x,y)j~F(x,y,z)= P(x,y,z)i+Q(x,y,z)j+R(x,y,z)k

The functions P, Q and R (if it is present) are called the scalarfunctions.

3 / 20

Vector Field

Example

A vector field on R2 is defined by~F(x,y)=yi+xj. Describe~F bysketching some vectors~F(x,y).

(x,y) ~F(x,y)

(1,0) 0,1(1,0) 0,1(0,1) 1,0

(0,1) 1,0(2,2) 2,2

(2,2) 2,2(2,2) 2,2(2,2) 2,2

4 / 20

3D Examples

If ~V (x,y,z) is the velocity vectorat any point (x,y,z), of a fluidflowing steadily along a pipe,

then ~V is a vector field, calledthe velocity field.

If a mass M is located at theorigin and another mass m hasposition vector~R= x,y,z thenthe gravitational force actingon this object is ~F(x,y,z) =mMG~R3

~R,

called the gravitational field.

5 / 20

Gradient Field

RECALL : The gradient of f :

f (x,y)= fx(x,y)i+ fy(x,y)j

f (x,y,z)= fx(x,y,z)i+ fy(x,y,z)j+ fz(x,y,z)k

Therefore, f is really a vector field and is called the gradient vectorfield.

Definition

A vector field~F is said to be conservative if it is a gradient vectorfield, that is, there is a function f such that f =~F .Such function f is called a potential function for~F .

6 / 20

Conservative Vector Field, an example

Example

The vector field~F(x,y)= (y32xy)i+ (3xy2x2+2)j is conservative.Find all potential functions of~F .

SOLUTION : Let f (x,y) be a potential function of~F . Then,

fx(x,y) = y32xyf (x,y) = xy3x2y+g(y)

fy(x,y) = 3xy2x2+g (y)3xy2x2+2 = 3xy2x2+g (y)

g (y) = 2g(y) = 2y+ c

Therefore,f (x,y)= xy3x2y+2y+ c

7 / 20

Conservative Vector Field, a counter-example

Example

Show that the vector field~F(x,y)= 2x+y,2xy is notconservative.

SOLUTION : Assume~F is consevative and let f (x,y) be a potentialfunction of~F . Then,

fx(x,y) = 2x+yf (x,y) = x2+xy+g(y)

fy(x,y) = x+g (y)2xy = x+g (y)

g (y) = xy

which is absurd !Hence, such function f does not exist, i.e.~F is not conservative.

8 / 20

Curl and Divergence

Definition

Let~F(x,y,z)= P(x,y,z)i+Q(x,y,z)j+R(x,y,z)k be a vector field on R3such that all partial derivatives of P, Q and R exist.

1 The divergence of~F , denoted div~F is the scalar field given by

div~F = Px(x,y,z)+Qy(x,y,z)+Rz(x,y,z)

2 The curl of~F , denoted curl~F , is the vector field given by

curl~F = [Ry(x,y,z)Qz(x,y,z)] i+ [Pz(x,y,z)Rx(x,y,z)] j+[Qx(x,y,z)Py(x,y,z)] k

9 / 20

Curl and Divergence

Define the differential operator (del) =

x,

y,

z

.

(i.e for f (x,y,z), f = fx, fy, fz)Then

div~F =~Fcurl~F =~F

Notice that the curl is defined only for a vector field on R3, so to findthe curl of a vector field on R2, we treat it as a vector field in R3 withR 0 (third component is the zero scalar function).

10 / 20

Curl and Divergence

Consider a velocity field~F in a fluid flow.

Particles near a point P(x,y,z) tend to rotateabout an axis in the direction of curl~F(x,y,z). Ifcurl~F =~0 at P, then~F is said to be irrotationalat P.

The divergence represents the net rate of change (with respect totime) of the mass of fluid flowing from P per unit volume, i.e.,div~F(x,y,z) measures the tendency of the fluid to diverge from P(sink/source). If div~F = 0 at P, then~F is said to be incompressible atP.

11 / 20

Curl and Divergence

Example

Determine the curl and divergence of~F = y2,2yz,xyz.SOLUTION

div~F = ~F=

x,

y,

z

y2,2yz,xyz

= (y2)

x+ (2yz)

y+ (xyz)

z= 2+xy

12 / 20

Curl and Divergence

Example

Determine the curl and divergence of~F = y2,2yz,xyz.SOLUTION

curl~F = ~F=

x,

y,

z

y2,2yz,xyz

=

i j kx

y

z

y2 2yz xyz

= xz+1,yz,2y

13 / 20

Some theorems

Theorem

Let~F(x,y,z)= P(x,y,z),Q(x,y,z),R(x,y,z) be a conservative vectorfield. If P, Q and R have continuous partial derivatives, thencurl~F =~0.PROOF. Suppose~F is conservative. Then there exists f (x,y,z) suchthat P = fx, Q= fy and R= fz.Also, since P, Q and R have continuous partial derivatives,

fxy = fyx, fyz = fzy, fxz = fzx.

Therefore,

curl~F =

i j kx

y

z

P Q R

=

i j kx

y

z

fx fy fz

= fzy fyz, fxz fzx, fyx fxy=~0

14 / 20

Some theorems

The converse of the previous theorem is not generally true. That is,if curl~F =~0, it does not follow immediately that~F is conservative.However, we have the following :

Corollary

If curl~F 6=~0, then~F is not conservative.

We have seen from a previous example that~F(x,y)= 2x+y,2xyis not conservative. Indeed,

curl~F =

i j kx

y

z

2x+y 2xy 0

= 0,0,1 6=~0

15 / 20

Some theorems

Theorem

Let~F(x,y,z)= P(x,y,z),Q(x,y,z),R(x,y,z). If P, Q and R havecontinuous second-order partial derivatives, then div(curl~F)= 0.

PROOF.

div(curl~F) = curl~F=

x,

y,

z

RyQz,PzRx,QxPy

= (RyQz)x

+ (PzRx)y

+ (QxPy)z

= RyxQzx+PzyRxy+QxzPyz= 0

since if P, Q and R have continuous partial derivatives, Pyz = Pzy,Qxz =Qzx and Rxy =Ryx.

16 / 20

Some theorems

Recall the following definitions

Definition1 A curve C is simple if it does not intersect itself.

2 A curve C is closed if its initial and terminal points coincide.

3 A region D is connected if any two points in D can beconnected by a path that lies entirely in D.

4 A connected region D is simply connected if every simpleclosed curve in D encloses points that are in D, that is, D hasno holes.

closed but not simple connected but not simply connected17 / 20

Some theorems

Theorem

If~F is a vector field defined on a simply-connected region DR3whose components have continuous partial derivatives, then~F isconservative if and only if curl~F =~0.

PROOF. See Calculus, Early Transcendentals Sec. 16.8.

Corollary

Let~F(x,y)= P(x,y),Q(x,y) be defined on a simply-connected regionsuch that P and Q have continuous partial derivatives. Then,~F isconservative if and only if Py(x,y)=Qx(x,y).

PROOF. Exercise.

18 / 20

Exercises

1 Determine whether or not~F is conservative. If it is, find allpotential functions for~F .

a. ~F(x,y)= ex siny,ex cosyb. ~F(x,y)= yex+ siny,ex+x cosyc. ~F(x,y)= y2,2xy+e3z,3ye3z

2 Find the curl and divergence of the vector field.a. ~F(x,y,z)= (xyz)i (x2y)kb. ~F(x,y,z)= lnx, lnxy, lnxyz

3 Assuming that the appropriate partial derivatives exist and arecontinuous, show that div(~F ~G)=~G curl~F ~F curl~G

4 Consider the vector field~F = y2,2yz,xyz.a. Find all points P(x,y,z) such that~F is incompressible.b. Show that~F can not be the curl of another vector field~G.

19 / 20

References

1 Stewart, J., Calculus, Early Transcendentals, 6 ed., ThomsonBrooks/Cole, 2008

2 Dawkins, P., Calculus 3, online notes available athttp ://tutorial.math.lamar.edu/

3 Gierach, M., Graber, H., Caruso, M., Remote Sensing of Environment,117(289U300) , 2012

20 / 20