week13 - national chiao tung...

CHAPTER 9 VECTOR CALCULUS-PART 2

WEN-BIN JIAN (簡紋濱)

DEPARTMENT OF ELECTROPHYSICS

NATIONAL CHIAO TUNG UNIVERSITY

OUTLINE

6. TANGENT PLANES AND NORMAL LINES

7. CURL AND DIVERGENCE

8. LINE INTEGRALS

9. INDEPENDENCE OF THE PATH

6. TANGENT PLANES AND NORMAL LINES

Example: Find the level curve passing and the gradient at for .

LC:

Example: Find the level surface of passing through .

LS:

Level Curves and Gradient

6. TANGENT PLANES AND NORMAL LINES

DEFINITION Tangent PlaneLet be a point on the surface of , where

, then the tangent plane is , where

.

Example: Please find the tangent plane and the normal line to the

surface of at the point .

The tangent plane is .

The normal line is .

Tangent Plane

10 010

10

0

10

10

0

10

7. CURL AND DIVERGENCE

Vector Functions – Vector FieldsTwo Variables Vector Functions – Vector Fields in 2D Space

For examples, ,

10 5 0 5 10

10

5

0

5

10

10 5 0 5 10

10

5

0

5

10

Three Variables Vector Functions – Vector Fields in 3D Space

Vector Fields

7. CURL AND DIVERGENCE

The gradient operation (on the scalar functions) is

and . Thus we define the

Del operator as .

The Curl operation (on the vector functions) is defined as .

The Divergence operation (on the vector functions) is defined as .

𝛻 �� = 𝚤𝜕

𝜕𝑥+ 𝚥

𝜕

𝜕𝑦+ 𝑘

𝜕

𝜕𝑧𝑓𝚤 + 𝑔𝚥 + ℎ𝑘 =

𝜕𝑓

𝜕𝑥+𝜕𝑔

𝜕𝑦+𝜕ℎ

𝜕𝑧

The Del Operator

7. CURL AND DIVERGENCE

Flux of a vector function across a surface (vector field):

Flux of a vector function in a small volume (vector field):

Concepts of The Divergence Calculation

7. CURL AND DIVERGENCE

Given a vector field in 3D space, , the net flux of the vector field through a small

cubic space is estimated as follows.

The flux in -coordinate through the small surface is

The net flux the small space is

The divergence of the vector field is the net flux of the vector field per unit volume, .

Concepts of The Divergence Calculation

7. CURL AND DIVERGENCE

For a curl-less vector field, like the electric field , you can choose a scalar potential because of the following operations.

For a divergence-less vector field, like the magnetic field , you can choose a vector potential because of the following operations.

Curl Less or Divergent Less Potential (Scalar or Vector Potential)

OUTLINE

6. TANGENT PLANES AND NORMAL LINES

7. CURL AND DIVERGENCE

8. LINE INTEGRALS

9. INDEPENDENCE OF THE PATH

8. LINE INTEGRALS

Let be a two-variable function, , defined on a region of the plane containing a smooth curve .

The line integral of along from A to B with respect to is

The line integral of along from A to B with respect to is

The line integral of along from A to B with respect to a curve is

Line Integrals on a 2D Plane

8. LINE INTEGRALS

If the curve is defined by an explicit function, that is representing the curve , the evaluations are done by the following ways.

𝑓 𝑥, 𝑦 𝑑𝑥 = 𝑓 𝑥, 𝑦 𝑥 𝑑𝑥

𝑓 𝑥, 𝑦 𝑑𝑦 = 𝑓 𝑥, 𝑦 𝑥 𝑦 𝑥 𝑑𝑥

𝑓 𝑥, 𝑦 𝑑𝑠 = 𝑓 𝑥, 𝑦 𝑥 1 + 𝑦 𝑥 

𝑑𝑥

Line Integrals on a 2D Plane

8. LINE INTEGRALS

If the curve is defined by an parametrical function, that is representing the curve , the evaluations are done by

the following ways.

Line Integrals on a 2D Plane

8. LINE INTEGRALS

Example 1: Evaluate (a) , (b) , and (c)

on the quarter circle defined by , , .

(a) /

= −256 cos 𝑡 sin 𝑡 𝑑𝑡/

= −256 sin 𝑡 𝑑 sin 𝑡/

Line Integrals on a 2D Plane

8. LINE INTEGRALS

Line Integrals on The Plane

Example 1: Evaluate (a) , (b) , and (c)

on the quarter circle defined by , , .

(b) /

/

/ /

  let

/

8. LINE INTEGRALS

Line Integrals on The Plane

Example 1: Evaluate (a) , (b) , and (c)

on the quarter circle defined by , , .

(c)  /

/

= 256 sin 𝑡 𝑑 sin 𝑡

/

/

8. LINE INTEGRALS

Example: Evaluate , where is given by ,

.

Example: Evaluate on the closed curve shown in the

figure.

𝑦 𝑑𝑥 − 𝑥 𝑑𝑦 = −72

5

Line Integrals on a 2D Plane

8. LINE INTEGRALS

Example: Evaluate , where is the helix

, , , .

Line Integrals on a 2D Plane

8. LINE INTEGRALS

 Circulation of :  

 for conservative forces.

Example: Find the work done by (a) and (b)

along the curve traced by , .

(a)  

(b)  

Line Integrals on a 2D Plane – Work Done by a Force

OUTLINE

6. TANGENT PLANES AND NORMAL LINES

7. CURL AND DIVERGENCE

8. LINE INTEGRALS

9. INDEPENDENCE OF THE PATH

9. INDEPENDENCE OF THE PATH

Example: Verify that the integral on paths of ,

, , and from to gives the same value.

(a)

(b)

(c)

(d)

Path Independent Integration Result

9. INDEPENDENCE OF THE PATH

DEFINITION Conservative Vector Field

A vector field in 2D or 3D space is conservative if can be written as the gradient of a scalar function . The function is called a potential function of .

Example: From the previous slide, we know that the integral

is independent of the path, the displacement in the

Cartesian coordinate is , then the integral can be

expressed as . The vector

field is said to be conservative if the integral is independent of the path.

Conservative Vector Fields

9. INDEPENDENCE OF THE PATH

THEOREM Fundamental TheoremSuppose is a path in an open region of the xy-plane and is defined by , . If

is a conservative vector field in and is a

potential function of then

.

If is a potential function of ,

�� 𝑥, 𝑦 = 𝛻𝜙 = 𝜙 𝚤 + 𝜙 𝚥, 𝑑𝑟 = 𝑑𝑥𝚤 + 𝑑𝑦𝚥

�� 𝑑𝑟 = 𝑑𝜙 = 𝜙 = 𝜙 𝐵 − 𝜙 𝐴

Conservative Vector Fields

9. INDEPENDENCE OF THE PATH

THEOREM Test for a Conservative Field

Suppose is a conservative vector field in an open region , and that and are continuous and have continuous first partial derivatives in . Then

, ,for all in . Conversely, if the equation hold

for all in a simply connected region , then is conservative in .

Conservative Vector Fields

9. INDEPENDENCE OF THE PATH

For a 3D conservative vector field and a piecewise-smooth space curve

, it shall satisfy the condition

if is conservative and are are continuous first partial derivatives in some open region in 3D space, then , ,

. Conversely, if the equation holds, is conservative.In addition, the curl of is a null vector. That is

.

Conservative of Mechanical Energy

In a conservative field , the law of conservation of mechanical energy holds.

Conservative Vector Fields

9. INDEPENDENCE OF THE PATH

Example: Determine whether the vector field is conservative.

Because , the vector field is conservative.

Test for a Conservative Vector Field

9. INDEPENDENCE OF THE PATH

Example: (a) Show that , where

is independent of the path between

and . (b) Find a potential function for . (c) Evaluate ,

,.

(a)

independent of the path

(b)  

𝜙 = 𝑄𝑑𝑦

= 𝑥𝑦 − 3𝑥 𝑦 − 𝑦 + 𝑔 𝑥

(c)

Test for a Conservative Vector Field

9. INDEPENDENCE OF THE PATH

Example: (a) Show that the line integral

is independent of the path between

(1,1,1) and (2,1,4). (b) Evaluate , ,

, ,.

(a)

the integration is independent of the path

Test for a Conservative Vector Field

9. INDEPENDENCE OF THE PATH

Example: (a) Show that the line integral

is independent of the path between

(1,1,1) and (2,1,4). (b) Evaluate , ,

, ,.

(b)  

𝜙 = 𝑄𝑑𝑦

+ 𝑔 𝑥, 𝑧 = 𝑥𝑦 + 𝑥𝑦𝑧 + 3𝑦𝑧 + 𝑔 𝑥, 𝑧

𝜙 = 𝑅𝑑𝑧

+ ℎ 𝑥, 𝑦 = 𝑥𝑦𝑧 + 3𝑦𝑧 − 𝑧 + ℎ 𝑥, 𝑦

∫ �� 𝑑𝑟, ,

, ,= 𝜙 2,1,4 − 𝜙 1,1,1

= 2 + 8 + 192 − 4 + 𝐶 − 1 + 1 + 3 − 1 + 𝐶 = 194

Test for a Conservative Vector Field