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CHAPTER 9 VECTOR CALCULUS-PART 2
WEN-BIN JIAN (簡紋濱)
DEPARTMENT OF ELECTROPHYSICS
NATIONAL CHIAO TUNG UNIVERSITY
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OUTLINE
6. TANGENT PLANES AND NORMAL LINES
7. CURL AND DIVERGENCE
8. LINE INTEGRALS
9. INDEPENDENCE OF THE PATH
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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
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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
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10 010
10
0
10
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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
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10 5 0 5 10
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0
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Three Variables Vector Functions – Vector Fields in 3D Space
Vector Fields
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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
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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
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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
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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)
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OUTLINE
6. TANGENT PLANES AND NORMAL LINES
7. CURL AND DIVERGENCE
8. LINE INTEGRALS
9. INDEPENDENCE OF THE PATH
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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
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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
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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
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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
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8. LINE INTEGRALS
Line Integrals on The Plane
Example 1: Evaluate (a) , (b) , and (c)
on the quarter circle defined by , , .
(b) /
/
/ /
let
/
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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
/
/
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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
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8. LINE INTEGRALS
Example: Evaluate , where is the helix
, , , .
Line Integrals on a 2D Plane
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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
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OUTLINE
6. TANGENT PLANES AND NORMAL LINES
7. CURL AND DIVERGENCE
8. LINE INTEGRALS
9. INDEPENDENCE OF THE PATH
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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
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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
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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
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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
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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
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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
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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
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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
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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