Chapter 2

Vector Analysis

¾ Spatial derivative of Scalar & Vector field: - Gradient of a Scalar field - Divergence of a vector field 2-5 Gradient of a Scalar Field ¾ Recall: given E, to find V Æ

(from vector to scalar)

¾ Now, what if given scalar field V, can we find a vector field E ??? Using gradient of a Scalar Field ¾ 2 equal-potential surfaces, V1, V1+dV ¾ 3 points: P1 at surface V1; P2, P3 at same surface V1+dV ¾ P1P2: along normal direction an ¾ P1P3: al · an = cos θ

2-5 Gradient of a Scalar Field ¾ Definition of gradient of a scalar field:

We define the vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar as the gradient of that scalar.

¾ Space rate of increase in the al direction: The projection of ∇V in the direction of al

also have

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2-5 Gradient of a Scalar Field ¾ In Cartesian coordinates:

¾ Vector differential operator:

¾ Vector differential operator in general orthogonal curvilinear coordinators :

Example 2-9: The electrostatic field intensity E is derivable as the negative gradient of an scalar electric potential V; that is, E = - ∇V. Determine E at the point (1, 1, 0) if (a) (b)

2-5 Gradient of a Scalar Field ¾ In other coordinates:

(see last page in the book)

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Chapter 2

Vector Analysis

2-6 Divergence of a Vector Field ¾ Recall Gauss’s law : ¾ For closed surface (sphere), net-electric-flux : ¾ Divergence of a vector A : Net outward flux of A per unit volume as the volume about the point tends to zero

¾ Div. is a measure of the strength of a flow source. ¾ A net outward flux of a vector A through a surface bounding a volume indicate the present of a flow source. ¾ Divergence in Cartesian : How do we find it ?

2-6 Divergence of a Vector Field ¾ We wish to derive ∇·A @ point P(x0, y0, z0) : - Consider a differential volume with side Δx, Δ y, Δ z - Center P(x0, y0, z0) - 6 surfaces (f, back, r, l, t, bottom)

¾ ∇·A in general orthogonal curvilinear coordinators :

¾ See the last page in textbook for ∇·A in Cylindrical and Spherical coordinators

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2-6 Divergence of a Vector Field

Example 2-10: p47 Find the divergence of the position vector to an arbitrary point.

Example 2-11: p47 The magnetic flux density B outside a very long current-carrying wire is circumferential and is inversely proportional to the distance to the axis of the wire. Find ∇·B

2-7 Divergence Theorem ¾ Divergence theorem :

Significance: It transformed the volume integral of the divergence of a vector field to a closed surface integral of the vector field, and vice versa

How to derive ? ¾ Definition of divergence: Div A at a point is defined as the net outward flux of A per unit volume as the volume about the point tends to zero

1) For small differential volume : 2) For any arbitrary volume V: (sum up)

3) Internal surfaces : cancel each other (due to opposite direction of dS) 4) External surfaces : net contribution, due to external surface S bounding the volume V Divergence theorem :

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Chapter 2

Vector Analysis

2-7 Divergence Theorem ¾ Significance of the divergence theorem: It converts a volume integral of the divergence of a vector to a closed surface integral of the vector, and vice versa. Example 2-12: Given a vector , verify the divergence theorem over a cube one unit on each side. The cube is situated in the first octant of the Cartesian coordinate system with one corner at the origin.

Chapter 2

Vector Analysis

2-8 Curl of a vector field ¾ There are two types of sources: Flow source :

div A is a measure of the strength of the flow source

Vortex source :

curl of A is a measure of the strength of the vortex source

¾ Vortex source causes a circulation of a vector field around it. ¾ Net circulation: ¾ Curl of a vector A (Definition):

Curl of a vector A is defined as max net circulation of vector A per unit area as area Æ 0

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2-8 Curl of a vector field

¾

¾ -

∇×A in Cartesian :

We wish to derive ∇×A @ point P(x0, y0, z0) : we can calculate x-components first (∇×A)x Consider a differential rectangular area Δ y, Δ z Center P(x0, y0, z0) 4 sides (1, 2, 3, 4)

¾ ∇×A in general orthogonal curvilinear coordinators :

¾ See the last page in textbook for ∇×A in Cylindrical and Spherical coordinators

2-8 Curl of a vector field Example 2-14: Given a vector OABO shown in fig. 2.22.

, find its circulation around the path

Example 2-15: Show that (a) (b)

if in cylindrical coordinates, where k is a constant, or in spherical coordinates, where f (R) is any function of the

radial distance R

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2-9 Stokes’s Theorem ¾ Stokes’s theorem : Significance: It transforms the surface integral of the curl of a vector field over a open surface to a closed line integral of the vector field over the contour bounding the surface, and vice versa

How to derive ? ¾ Definition of curl of a vector: Curl of a vector A is defined as max net circulation of vector A per unit area as area Æ 0

1) For small differential area : 2) For any arbitrary surface S : (sum up)

3) Internal contour : cancel each other ( due to opposite directions of dl ) 4) External contour : net contribution, due to external contour C bounding the entire area S

C

Stokes’s theorem :

2-9 Stokes’s Theorem ¾ Significance of Stokes’s theorem: It transforms the surface integral of the curl of a vector field over a open surface to a closed line integral of the vector field over the contour bounding the surface, and vice versa

¾ Special case 1: Surface integral of ∇×A is carried over closed surface (3D)

¾ Special case 2: 3D Æ 2D, such as 2D disk ¾ What about the direction of dS and dl ? The relative directions of dl and dS (dan) follow the right-hand rule !

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2-9 Stokes’s Theorem

2-10

Two Null Identities

¾ Two identities involving “del” operation:

¾ Identities I :

Identities I

Identities II

The curl of the gradient of any scalar field is identically zero.

How to derive ? By using Stokes’s theorem

Converse statement : If a vector field is curl-free, then it is a conservative (or irrotational) field, and can be expressed as the gradient of a scalar field. Example: If

, then we can define electric scalar potential V:

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2-10

Two Null Identities

¾ Identities II :

The divergence of the curl of any vector field is identically zero.

How to derive ? By using Divergence theorem

Note: The right side of above equation, is a closed surface. We may split it into 2 open surfaces so as to use Stokes’s theorem.

Converse statement : If a vector field is divergenceless, then it is solenoidal field and can be expressed as the curl of another vector field. Example: If

, then we can define magnetic vector potential A:

Chapter 2

Vector Analysis

Summary: ¾ Reviewed the basic rules of vector addition and subtraction, and of products of vectors ¾ Explained the properties of Cartesian, cylindrical, and spherical coordinate systems ¾ Introduced the differential del ( ) operator, and defined the gradient of a scalar field, and the divergence and the curl of a vector field ¾ Presented the divergence theorem that transformed the volume integral of the divergence of a vector field to a closed surface integral of the vector field, and vice versa ¾ Presented the Stokes’s theorem that transforms the surface integral of the curl of a vector field to a closed line integral of the vector field, and vice versa ¾ Introduced two important null identities in vector field

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