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


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


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

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0

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0

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

For examples, ,

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10

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0

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Three Variables Vector Functions – Vector Fields in 3D Space

Vector Fields


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


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


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


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



8. LINE INTEGRALS


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


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.


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/


8. LINE INTEGRALS

Line Integrals on The Plane



(b) /

/

/ /

let

/

8. LINE INTEGRALS

Line Integrals on The Plane



(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


8. LINE INTEGRALS

Example: Evaluate , where is the helix

, , , .


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



8. LINE INTEGRALS



Example: Verify that the integral on paths of ,

, , and from to gives the same value.

(a)

(b)

(c)

(d)

Path Independent Integration Result


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


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



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 .



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.



Example: Determine whether the vector field is conservative.

Because , the vector field is conservative.

Test for a Conservative Vector Field


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)



Example: (a) Show that the line integral


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

, ,.

(a)

the integration is independent of the path



Example: (a) Show that the line integral


(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


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