Lecture 12: Vector Calculus II

1. Key pointsVector fieldsField Lines/Flow LinesDivergenceCurl

Maple commandsVectorCalculus packageStudent[VectorCalculus] packagePhysics[Vector] packageVectorFieldDivergenceCurlFlowLine

2. Vector Fields and Field Lines (Flow Lines)A vector field has a vector value at each point of space and expressed as a vector-valued function. In the Cartesian coordinate, it is written as

Vector fields in physics:

( ),

electric field ( ),

magnetic field ( ),

vector potential ( ),

current ( ).

The gradient of a scalar field is a vector field. See last lecture.

A field line is the curve where the field at every point on the curve is tangent to the curve.

Example field line

=

Example - Visualizing vector fields

=

Maple vector field tutor

(2.4.1.1)(2.4.1.1)

Exercise

Visualize the force field of a 3D simple harmonic oscillator: . Plot

also the force field in the xy plane (z=0).

Answer

=

3. Divergence

The dot product between a del operator and a vector field is called the divergence of the vector field

div ( ) =

Since the output of dot product is scalar, the divergence of a vector field is a scalar field.

Divergence in physics:

Maxwell's equations: and .

continuity equation: 0

Example - Visualizing divergence

For simplicity, we consider two dimensional space first.

=

=

Note that the divergence is negative when the arrows become smaller along the flow and positive when the arrows grow along the flow.

ExerciseFind the divergence of the following vector fields.

(1) .(2) .

Solution

(1) r = = 3(2) V, F = =

4. CurlThe cross product of a del operator and a vector field is called the curl of the vector field.

curl ( ) = .

Since the output of cross product is vector, the curl of a vector field is another vector field.

Example - Visualizing curl

=

=

The vector field shows a rotation of arrows and the curl shows vectors perpendicular to the rotation. The curl extracts information about rotation in vector fields.

Examples in Physics

Force is conservative if .

Maxwell's equations: and .

Vector potential: .

ExerciseCalculate the curl of the following vector fields.

(1) ( ) = .

(2) .

(3)

.

Solution

(1) = = .

(2) = = .

(3) = = .

5. Combination of grad, div, and curl(1) div+grad = (Laplacian)(2) curl+grad = for any scalar field.(3) div+curl F = for any vector field.(4) curl+curl F = grad+div F - Laplacian F(5) div (4 F) = V, (4F) = 4(V, F)+F (grad 4)(6) curl (4 F) = V# (4F) F)+(V4)×F = 4 (curl F) + (grad 4)×F.(7) div (F G) = V, (F G) = F G F G = F G F G(8) curl (F G) = V# (F G) = (F G+F(V, G G F G(V, F)(9) grad (F G) = F G)+(F G +G F)+(G F(10) div [(grad )×(grad 4)] = V, (Vs# V4) = 0 for any scalar fields and 4.

div+grad =

=

= = simplify

curl+grad = =

=

div+curl F =

= 0 = 0

curl+curl F = grad+div F - Laplacian F

=

6. Maple Physics[Vectors] Package

When Maple Physics Vectors Package is loaded, many common expression used in physics are automatically assumed. For example, you don't have to set coordinates if you use common expression. See conventions.

Concerning the coordinates, the conventions are:

= Cartesian coordinates,

= cylindrical coordinates,

= spherical coordinates

= Cartesian unit vectors,

= cylindrical unit vectors,

= spherical unit vectors

Gradient

=

=

Laplacian

=

=

Divergence

=

=

Curl

(6.2)(6.2)

(6.1)(6.1)

(6.3)(6.3)

=

=

Change of Basis

Homework: Due 10/16, 11am

12.1(1) Find the divergence of the following vector fields by hand and confirm them with Maple.

(a)

(b)

(2) For a vector field , show that .

12.2Calculate the curl of the following vector fields. Visually compare the original field and its curl.

(1)

(2) .

12.3

In E&M class, magnetic field can be obtained from vector potential using a formula

. Find the magnetic field from the following vector potential.

where and are constants.

12.4If a potential field depends only on radial coordinate in the spherical coordinates as . The

corresponding force is given by . Show the following properties:

(1)

(2)

12.5

Electric field is related to electrostatic potential by . Show that the electric field line is perpendicular to the equipotential surface where is a constant.