12 Vector Fields - Handout

Vector Fields

Math 55 - Elementary Analysis III

Institute of MathematicsUniversity of the Philippines

Diliman

Math 55 Vector Fields 1/ 20

Vector Fields

Definition

A vector field on R2 (or R3) is a function ~F that assigns toeach (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)) with initial point (x, y) (or (x, y, z)) are calledflowlines 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.


Example

The following shows air velocity vectors that indicate the wind speedand direction at points above the surface elevation in Philippine areaon March 16, 2009 00:00 UTC.

To each point in the air we associate a vector corresponding to the

velocity of the wind.


Vector Field

Example

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


3D Examples

If ~V (x, y, z) is the velocityvector at any point (x, y, z), ofa fluid flowing steadily along apipe,

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

If a mass M is located atthe origin and another massm has position vector ~R =x, y, z then the gravitationalforce acting on this object is

~F (x, y, z) = mMG~R3~R,

called the gravitational field.


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 gradientvector field.

Definition

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


Conservative Vector Field, an example

Example

The vector field ~F (x, y) = (y3 2xy)i + (3xy2 x2 + 2)j isconservative. Find all potential functions of ~F .

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

fx(x, y) = y3 2xy

f(x, y) = xy3 x2y + g(y)fy(x, y) = 3xy

2 x2 + g(y)3xy2 x2 + 2 = 3xy2 x2 + g(y)

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

Therefore,f(x, y) = xy3 x2y + 2y + c


Conservative Vector Field, a counter-example

Example

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

Solution: Assume ~F is consevative and let f(x, y) be apotential function of ~F . Then,

fx(x, y) = 2x + y

f(x, y) = x2 + xy + g(y)

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

2x y = x + g(y)g(y) = x y

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


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 vectorfield on R3 such that all partial derivatives of P , Q and R exist.

1 The divergence of ~F , denoted div~F is the scalar fieldgiven 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


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 tofind the curl of a vector field on R2, we treat it as a vector fieldin R3 with R 0 (third component is the zero scalar function).


Curl and Divergence

Consider a velocity field ~F in a fluid flow.

Particles near a point P (x, y, z) tend torotate about an axis in the direction ofcurl~F (x, y, z). If curl~F = ~0 at P , then ~Fis said to be irrotational at P .

The divergence represents the net rate of change (with respectto time) 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 divergefrom P (sink/source). If div~F = 0 at P , then ~F is said to beincompressible at P .


Curl and Divergence

Example

Determine the curl and divergence of ~F =y2, 2y z, xyz.

Solution

div~F = ~F=

x,

y,

z

y2, 2y z, xyz

=(y2)

x+(2y z)

y+(xyz)

z= 2 + xy


Curl and Divergence

Example

Determine the curl and divergence of ~F =y2, 2y z, xyz.

Solution

curl~F = ~F=

x,

y,

z

y2, 2y z, xyz

=

i j kx

y

z

y2 2y z xyz

= xz + 1,yz,2y


Some theorems

Theorem

Let ~F (x, y, z) = P (x, y, z), Q(x, y, z), R(x, y, z) be aconservative vector field. If P , Q and R have continuous partialderivatives, then curl~F = ~0.

Proof. Suppose ~F is conservative. Then there exists f(x, y, z)such that 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


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 isconservative. 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, 2x y is not conservative. Indeed,

curl~F =

i j kx

y

z

2x + y 2x y 0

= 0, 0, 1 6= ~0


Some theorems

Theorem

Let ~F (x, y, z) = P (x, y, z), Q(x, y, z), R(x, y, z). If P , Q and Rhave continuous second-order partial derivatives, thendiv(curl~F ) = 0.

Proof.

div(curl~F ) = curl~F=

x,

y,

z

Ry Qz, Pz Rx, Qx Py

=(Ry Qz)

x+(Pz Rx)

y+(Qx Py)

z= Ryx Qzx + Pzy Rxy + Qxz Pyz= 0

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


Some theorems

Definition

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

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

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 everysimple closed curve in D encloses points that are in D, thatis, D has no holes.

closed but not simple connected but not simply connected


Some theorems

Theorem

If ~F is a vector field defined on a simply-connected regionD R3 whose components have continuous partial derivatives,then ~F is conservative 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-connectedregion such that P and Q have continuous partial derivatives.Then, ~F is conservative if and only if Py(x, y) = Qx(x, y).

Proof. Exercise.


Exercises

1 Determine whether or not ~F is conservative. If it is, find all

potential functions for ~F .

a. ~F (x, y) = ex sin y, ex cos yb. ~F (x, y) = yex + sin y, ex + x cos yc. ~F (x, y, z) =

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 andare continuous, show thatdiv(~F ~G) = ~G curl~F ~F curl~G

4 Consider the vector field ~F =y2, 2y z, 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.


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 ofEnvironment, 117(289300) , 2012


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12 Vector Fields - Handout