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CHAPTER 9 VECTOR CALCULUS- PART 4 WEN-BIN JIAN (簡紋濱) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY
CHAPTER 9 VECTOR CALCULUS-PART 4
WEN-BIN JIAN (簡紋濱)
DEPARTMENT OF ELECTROPHYSICS
NATIONAL CHIAO TUNG UNIVERSITY
OUTLINE
14.STOKE’S THEOREM (CURL THEOREM)
15.TRIPLE INTEGRALS
16.DIVERGENCE THEOREM
The Vector Format of Green’s Theorem
14. STOKE’S THEOREM
If is a 2D vector field,
Green’s Theorem can be written in vector form:
Green’s Theorem in 3D Coordinate Space
14. STOKE’S THEOREM
Green’s theorem in 3D space is called Stokes’ theorem after the Irish mathematical physicist George G. Stokes (1819-1903).
THEOREM Stokes’ Theorem
Let S be a piecewise-smooth orientable surface bounded by a apiecewise-smooth simple closed curve C. Let be a vector field for which are continuous and have first partial derivatives in a region of 3D space containing S. If C is traversed in
the positive direction, then . Where
n is a unit vector to S in the direction of the orientation of S.
14. STOKE’S THEOREM
Stoke’s Theorem
Assume a surface function of ,
𝛻 × �⃗� 𝑁 =
Proof of Stoke’s Theorem
14. STOKE’S THEOREM
�⃗� 𝑑𝑟 = 𝑓𝑑𝑥 + 𝑔𝑑𝑦 + ℎ𝑑𝑧 , 𝑧 = 𝑢 𝑥, 𝑦
�⃗� 𝑑𝑟 = 𝑓𝑑𝑥 + 𝑔𝑑𝑦 + ℎ𝑢 𝑑𝑥 + ℎ𝑢 𝑑𝑦
= 𝑓 𝑥, 𝑦, 𝑧 + ℎ 𝑥, 𝑦, 𝑧 𝑢 𝑥, 𝑦 𝑑𝑥 + 𝑔 + ℎ𝑢 𝑑𝑦
Proof of Stoke’s Theorem
Example: Let S be the part of the cylinder for , . Verify Stoke’s theorem for the vector field . Assume S is oriented upward.
14. STOKE’S THEOREM
along
along the y direction
along
along the y direction�⃗� 𝑑𝑟 = −2
Applications of Stoke’s Theorem
14. STOKE’S THEOREM
𝛻 × �⃗� 𝑁𝑑𝑠 = −2𝑥𝑦 − 𝑥 𝑑𝑥𝑑𝑦
= −2
Example: Let S be the part of the cylinder for , . Verify Stoke’s theorem for the vector field . Assume S is oriented upward.
Applications of Stoke’s Theorem
Example:
, where C is the trace of the cylinder
in the plane . Orient C counterclockwise in the positive z axis.
14. STOKE’S THEOREM
𝛻 × �⃗� 𝑁𝑑𝑠 = 2𝑑𝑥𝑑𝑦
= 2𝑟𝑑𝜃𝑑𝑟 = 2𝜋
Applications of Stoke’s Theorem
14. STOKE’S THEOREM
The circulation of vector field
The vector field tends to turn around the curve
The curl of is a measure of the rotational strength of per unit area.
Concept of Stoke’s (Curl) Theorem
OUTLINE
14.STOKE’S THEOREM (CURL THEOREM)
15.TRIPLE INTEGRALS
16.DIVERGENCE THEOREM
15. TRIPLE INTEGRALS
Let be a scalar function of three variables defined over a closed region D in the 3D space. The triple integral of over D is
given by .
The Calculation:
𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = 𝑓 𝑥, 𝑦, 𝑧 𝑑𝑧𝑑𝑦𝑑𝑥,
,
𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = 𝑓 𝑥, 𝑦, 𝑧 𝑑𝑥𝑑𝑧𝑑𝑦,
,
𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = 𝑓 𝑥, 𝑦, 𝑧 𝑑𝑦𝑑𝑥𝑑𝑧,
,
The Triple Integrals – Volume Calculation
15. TRIPLE INTEGRALS
Estimation of Volume:
Estimation of Mass:
Estimation of The Center of Mass: ,
,
Estimation of Moment of Inertia: ∟
Radius of Gyration:
Application of The Triple Integrals
15. TRIPLE INTEGRALS
Example: Find the volume of the solid in the first octant bounded by the graphs of , , and .
/
according to the boundary, x and z are dependent on y
Application of The Triple Integrals
15. TRIPLE INTEGRALS
Example: Change the order of integration from
to
x is in the range from 0 to 6
x is fixed, y is in the range from y=0 to y=4-2x/3(between the two lines of y=0 and 2x+3y=12)
x and y are fixed, z is in the range fromz=0 to z=3-x/2-3y/4 (2x+3y+4z=12)
𝑓 𝑥, 𝑦, 𝑧 𝑑𝑦𝑑𝑥𝑑𝑧
Application of The Triple Integrals
15. TRIPLE INTEGRALS
It is just the polar coordinate combined with the z-axis.
The variables are .
Change the variables from to .
Change the variables from to .
The triple integral:
𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = 𝑓 𝑥, 𝑦, 𝑧 𝑟𝑑𝜃 𝑑𝑟 𝑑𝑧
Cylindrical Coordinate
15. TRIPLE INTEGRALS
Example: Convert in cylindrical coordinate to rectangular coordinate.
Example: A solid in the first octant has the shape determined by the
graph of the cone and the plane . Find the center of mass if the density is given by .
Applications of Cylindrical Coordinate
15. TRIPLE INTEGRALS
Example: A solid in the first octant has the shape determined by the graph of the cone and the plane . Find the center of mass if the density is given by .
cylindrical symmetry -> use cylindrical coordinate
/
/
/
Applications of Cylindrical Coordinate
15. TRIPLE INTEGRALS
The variables are .
Change the variables from to .
Change the variables from to .
The triple integral:
𝑓 𝑥, 𝑦, 𝑧 𝑑𝑉 = 𝑓 𝑥, 𝑦, 𝑧 𝑟 sin 𝜃 𝑑𝜙 𝑟𝑑𝜃 𝑑𝑟
Spherical Coordinate
15. TRIPLE INTEGRALS
Example: Find the moment of inertia about the z-axis of the homogeneous solid bounded between the spheres and , .
assume a uniform density , the mass is
∟
Applications of Spherical Coordinate
OUTLINE
14.STOKE’S THEOREM (CURL THEOREM)
15.TRIPLE INTEGRALS
16.DIVERGENCE THEOREM
16. DIVERGENCE THEOREM
If is a 2D vector field, Green’s theorem gives
�⃗� 𝑑𝑟
= 𝑃 𝑥, 𝑦 𝑑𝑥 + 𝑄 𝑥, 𝑦 𝑑𝑦 = −𝜕𝑃 𝑥, 𝑦
𝜕𝑦+𝜕𝑄 𝑥, 𝑦
𝜕𝑥𝑑𝑥𝑑𝑦
In this case is along the curve of line integral
rotate the for to be
Another vector form of Green’s theorem
∮ �⃗� 𝑑𝑟
= ∮ 𝑃 𝑥, 𝑦 𝑑𝑦 − 𝑄 𝑥, 𝑦 𝑑𝑥
=𝜕𝑃 𝑥, 𝑦
𝜕𝑥+𝜕𝑄 𝑥, 𝑦
𝜕𝑦𝑑𝑥𝑑𝑦 = 𝛻 �⃗� 𝑑𝑥𝑑𝑦
Extended from The Green’s Theorem
16. DIVERGENCE THEOREM
�⃗� 𝑑𝑟
= 𝑃 𝑥, 𝑦 𝑑𝑦 − 𝑄 𝑥, 𝑦 𝑑𝑥 = 𝛻 �⃗� 𝑑𝑥𝑑𝑦
Generalize to 3D space, take the rotated as the surface normal of the plane
Let D be a closed and bounded region in 3D space with a piecewise-smooth boundary S that is oriented outward. Let
be a vector field for which are continuous and their first partial derivatives are continuous in a region of 3D space D. Then
Divergence Theorem
Divergence Theorem – Interpretation Using The Flux of Vector Fields
Divergence is a measure of flux per unit volume of vector field.
16. DIVERGENCE THEOREM
∯ 𝑅 𝑥, 𝑦, 𝑧 𝑘 𝑛 𝑑𝑠 =?∭, ,
𝑑𝑉
𝜕𝑅 𝑥, 𝑦, 𝑧
𝜕𝑧𝑑𝑧𝑑𝑥𝑑𝑦
∯ 𝑅 𝑥, 𝑦, 𝑧 𝑘 𝑛 𝑑𝑠 = ∬ 𝑅 𝑥, 𝑦, 𝑧 𝑘 𝑛 𝑑𝑠
+∬ 𝑅 𝑥, 𝑦, 𝑧 𝑘 𝑛 𝑑𝑠+∬ 𝑅 𝑥, 𝑦, 𝑧 𝑘 𝑛 𝑑𝑠
∬ 𝑅 𝑥, 𝑦, 𝑧 𝑘 𝑛 𝑑𝑠 = 0 For S1: ̂ ̂
−𝑅 𝑥, 𝑦, 𝑧
1 +𝜕𝑧𝜕𝑥
+𝜕𝑧𝜕𝑦
1 +𝜕𝑧
𝜕𝑥+
𝜕𝑧
𝜕𝑦
𝑑𝑥𝑑𝑦 = − 𝑅 𝑥, 𝑦, 𝑧 𝑑𝑥𝑑𝑦
Proof of The Divergence Theorem
16. DIVERGENCE THEOREM
∯ 𝑅 𝑥, 𝑦, 𝑧 𝑘 𝑛 𝑑𝑠 =∭, ,
𝑑𝑉
For S2: ̂ ̂
𝑅 𝑥, 𝑦, 𝑧
1 +𝜕𝑧𝜕𝑥
+𝜕𝑧𝜕𝑦
1 +𝜕𝑧
𝜕𝑥+
𝜕𝑧
𝜕𝑦
𝑑𝑥𝑑𝑦 = 𝑅 𝑥, 𝑦, 𝑧 𝑑𝑥𝑑𝑦
∯ 𝑅 𝑥, 𝑦, 𝑧 𝑘 𝑛 𝑑𝑠 = ∬ 𝑅 𝑥, 𝑦, 𝑧 − 𝑅 𝑥, 𝑦, 𝑧
𝑑𝑥𝑑𝑦
Proof of The Divergence Theorem
16. DIVERGENCE THEOREM
Example: Let D be the region bounded by the hemisphere , , and the plane . Verify the divergence
theorem if .
To get , we need
To get , we need
Applications of The Divergence Theorem
16. DIVERGENCE THEOREM
on spherical surface , on the plane
∯ �⃗� 𝑛 𝑑𝑠 = ∬ 𝑑𝑥𝑑𝑦
= ∫ ∫ 𝑟𝑑𝜃𝑑𝑟 =
18𝜋 − 9 − 𝑟
= 54𝜋
∭ 𝛻 �⃗� 𝑑𝑉 =∭ 3𝑑𝑉 = 3 3 = 54𝜋
Example: Let D be the region bounded by the hemisphere , , and the plane . Verify the divergence
theorem if .
Applications of The Divergence Theorem
16. DIVERGENCE THEOREM
Example: If , evaluate where S is
the unit cube defined by .
∯ �⃗� 𝑛 𝑑𝑠 = ∫ ∫ ∫ 𝑦 + 2𝑦𝑧 + 3𝑧 𝑑𝑥𝑑𝑦𝑑𝑧
Applications of The Divergence Theorem
16. DIVERGENCE THEOREM
Divergence of is the net flux per unit volume.
Continuity Equation:
If the volume flow is a vector field (must be the particle velocity), where the unit is length per unit time, then the mass flow is expressed by .
The continuity equation is only a mass conservation law:
The positive out flowing of the mass corresponds to the reduction (negative value) of the mass in the bounded volume.
Interpretation of The Divergence Theorem
