Vector Calculus Dr. D. Sukumar January 31, 2014
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Vector Calculus
Dr. D. Sukumar
January 31, 2014
Green’s TheoremTangent form or Ciculation-Curl form
‰
c
Mdx + Ndy =
¨
R
(
∂N
∂x− ∂M
∂y
)
dA
Green’s TheoremTangent form or Ciculation-Curl form
‰
c
Mdx + Ndy =
¨
R
(
∂N
∂x− ∂M
∂y
)
dA
‰
C
F · dr =¨
R
(∇× F ) · k dA
◮ C is a simple, closed, smooth curve in counterclockwisedirection
◮ R is the region enclosed by C
◮ dA is area element
◮ dr is tangential length
Stokes Theorem
The circulation of F = M i+ Nj+ Pk around the boundary C
of an oriented surface S in the direction counterclockwise withrespect to the surface’s unit normal vector n equals theintegral of ∇× F · n over S
‰
C
F · dr =¨
S
∇× F · n dσ
◮ C is a simple,closed, smooth curve consideredcounterclockwise direction
Stokes Theorem
The circulation of F = M i+ Nj+ Pk around the boundary C
of an oriented surface S in the direction counterclockwise withrespect to the surface’s unit normal vector n equals theintegral of ∇× F · n over S
‰
C
F · dr =¨
S
∇× F · n dσ
◮ C is a simple,closed, smooth curve consideredcounterclockwise direction
◮ S is a surface (oriented) with boundary C
Stokes Theorem
The circulation of F = M i+ Nj+ Pk around the boundary C
of an oriented surface S in the direction counterclockwise withrespect to the surface’s unit normal vector n equals theintegral of ∇× F · n over S
‰
C
F · dr =¨
S
∇× F · n dσ
◮ C is a simple,closed, smooth curve consideredcounterclockwise direction
◮ S is a surface (oriented) with boundary C
◮ dσ is surface area element
Stokes Theorem
The circulation of F = M i+ Nj+ Pk around the boundary C
of an oriented surface S in the direction counterclockwise withrespect to the surface’s unit normal vector n equals theintegral of ∇× F · n over S
‰
C
F · dr =¨
S
∇× F · n dσ
◮ C is a simple,closed, smooth curve consideredcounterclockwise direction
◮ S is a surface (oriented) with boundary C
◮ dσ is surface area element
◮ dr is tangential length
Use Stoke’s Theorem to calculate the circulation of the FieldF = x2i + 2xj + z2k around the curve C : The ellipse4x2 + y2 = 4 in the xy plane counter clockwise when viewedfrom above.
Use Stoke’s Theorem to calculate the circulation of the FieldF = x2i + 2xj + z2k around the curve C : The ellipse4x2 + y2 = 4 in the xy plane counter clockwise when viewedfrom above.The surface S is 4x2 + y2 = 4 ie f = 4x2 + y2 = 4
Use Stoke’s Theorem to calculate the circulation of the FieldF = x2i + 2xj + z2k around the curve C : The ellipse4x2 + y2 = 4 in the xy plane counter clockwise when viewedfrom above.The surface S is 4x2 + y2 = 4 ie f = 4x2 + y2 = 4
∇× F =
∣
∣
∣
∣
∣
∣
i j k∂
∂x∂
∂y∂
∂z
x2 2x z2
∣
∣
∣
∣
∣
∣
(∇× F ) · k = 2
Use Stoke’s Theorem to calculate the circulation of the FieldF = x2i + 2xj + z2k around the curve C : The ellipse4x2 + y2 = 4 in the xy plane counter clockwise when viewedfrom above.The surface S is 4x2 + y2 = 4 ie f = 4x2 + y2 = 4
∇× F =
∣
∣
∣
∣
∣
∣
i j k∂
∂x∂
∂y∂
∂z
x2 2x z2
∣
∣
∣
∣
∣
∣
(∇× F ) · k = 2
Circulation
‰
c
F · dr
Use Stoke’s Theorem to calculate the circulation of the FieldF = x2i + 2xj + z2k around the curve C : The ellipse4x2 + y2 = 4 in the xy plane counter clockwise when viewedfrom above.The surface S is 4x2 + y2 = 4 ie f = 4x2 + y2 = 4
∇× F =
∣
∣
∣
∣
∣
∣
i j k∂
∂x∂
∂y∂
∂z
x2 2x z2
∣
∣
∣
∣
∣
∣
(∇× F ) · k = 2
Circulation
‰
c
F · dr =ˆ 2
0
ˆ
√4−y2
2
−√
4−y2
2
2dxdy = 2
ˆ 2
0
√
4− y2dy
Use Stoke’s Theorem to calculate the circulation of the FieldF = x2i + 2xj + z2k around the curve C : The ellipse4x2 + y2 = 4 in the xy plane counter clockwise when viewedfrom above.The surface S is 4x2 + y2 = 4 ie f = 4x2 + y2 = 4
∇× F =
∣
∣
∣
∣
∣
∣
i j k∂
∂x∂
∂y∂
∂z
x2 2x z2
∣
∣
∣
∣
∣
∣
(∇× F ) · k = 2
Circulation
‰
c
F · dr =ˆ 2
0
ˆ
√4−y2
2
−√
4−y2
2
2dxdy = 2
ˆ 2
0
√
4− y2dy
=2
(
y
2
√
4− y2 +4
2sin−1 y
2
)2
0
Use Stoke’s Theorem to calculate the circulation of the FieldF = x2i + 2xj + z2k around the curve C : The ellipse4x2 + y2 = 4 in the xy plane counter clockwise when viewedfrom above.The surface S is 4x2 + y2 = 4 ie f = 4x2 + y2 = 4
∇× F =
∣
∣
∣
∣
∣
∣
i j k∂
∂x∂
∂y∂
∂z
x2 2x z2
∣
∣
∣
∣
∣
∣
(∇× F ) · k = 2
Circulation
‰
c
F · dr =ˆ 2
0
ˆ
√4−y2
2
−√
4−y2
2
2dxdy = 2
ˆ 2
0
√
4− y2dy
=2
(
y
2
√
4− y2 +4
2sin−1 y
2
)2
0
=4(
π
2− 0
)
= 2π
ExerciseStoke’s Theorem
Use Stoke’s theorem to calculate the flux of the curl of thefield F across the surface S in the direction of the outwardunit normal n.
1. F = 2zi + 3xj + 5ykS : z + x2 + y2 = 4 12π
2. F = 2zi + 3xj + 5ykS : r(r , θ) = (r cos θ)i + (r sin θ)j + (4− r2)k0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π
3. F = x2yi + 2y3zj + 3zkS : r(r , θ) = (r cos θ)i + (r sin θ)j + rk
0 ≤ r ≤ 1, 0 ≤ θ ≤ 2π−π
4
Green’s Theorem(Normal form or Flux-Divergence form)
˛
C
Mdy − Ndx =
¨
R
(
∂M
∂x+
∂N
∂y
)
dA
Green’s Theorem(Normal form or Flux-Divergence form)
˛
C
Mdy − Ndx =
¨
R
(
∂M
∂x+
∂N
∂y
)
dA
˛
C
F · n ds =¨
S
∇ · F dA
◮ C is a simple, closed, smooth curve
◮ R is the region enclosed by C
◮ dA is area element
◮ ds is length element
˛
C
F · n ds =¨
S
∇ · F dA
˛
C
F · n ds =¨
S
∇ · F dA
¨
S
F · n dσ =
˚
D
∇ · F dV
◮ S is a simple, closed, oriented surface.
◮ D is solid regin bounded by S
◮ dσ surface area element
◮ dV is volume element
The Divergence Theorem
The flux of a vector field F = M i+Nj+ Pk across a closedoriented surface S in the direction of the surface’s outwardunit normal field n equals the integral of ∇ · F (divergence ofF ) over the region D enclosed by the surface:
¨
S
F · n dσ =
˚
D
∇ · F dV .
F = yi + xyi − zkD : The region inside the solid cylinder x2 + y2 ≤ 4 betweenthe plane z = 0 and the parabolaid z = x2 + y2
∇ · F = 0+ x − 1 = x − 1˚
D
∇ · F dV
F = yi + xyi − zkD : The region inside the solid cylinder x2 + y2 ≤ 4 betweenthe plane z = 0 and the parabolaid z = x2 + y2
∇ · F = 0+ x − 1 = x − 1˚
D
∇ · F dV =
ˆ
2
0
ˆ
√4−x2
−√4−x2
ˆ x2+y2
0
(x − 1)dzdydx
=
ˆ
2
0
ˆ
√4−x2
−√4−x2
(x − 1)(x2 + y2)dydx
=
ˆ
2
0
(x − 1)[x2y +y3
3]√4−x2
−√4−x2
=
ˆ
2
0
(x − 1)(2x2√
4− x2 +2
3(4− x)2
√
4− x2)dx
=1
3
ˆ
2
0
(x − 1)√
4− x2[6x2 + 2(16− 8x + 8x2)]dx
=1
3
ˆ
2
0
(x − 1)√
4− x2[8x2 − 8x + 16]dx
= −16π
ExerciseDivergence theorem
Use divergence theorem to calculate outward flux
1. F = (y − x)i+ (z − y )j+ (y − x)kD :The cube bounded by the planes x ± 1, y ± 1 andz ± 1. −16
2. F = x2i− 2xy j+ 3xzkD :The region cut from the first octant by the spherex2 + y2 + z2 = 4 3π
◮ F is conservative, F is irrotational=⇒ Ciruculation= 0
◮ F is incompressible, ∇.F is 0 =⇒ Flux= 0
Fundamental Theorem of Calculus
ˆ
[a,b]
df
dxdx = f (b)− f (a)
Fundamental Theorem of Calculus
ˆ
[a,b]
df
dxdx = f (b)− f (a)
Let F = f (x)i
ˆ
[a,b]
df
dxdx = f (b)− f (a)
= f (b)i · i + f (a)i · −i
Fundamental Theorem of Calculus
ˆ
[a,b]
df
dxdx = f (b)− f (a)
Let F = f (x)i
ˆ
[a,b]
df
dxdx = f (b)− f (a)
= f (b)i · i + f (a)i · −i
= F (b) · n+ F (a) · n
Fundamental Theorem of Calculus
ˆ
[a,b]
df
dxdx = f (b)− f (a)
Let F = f (x)i
ˆ
[a,b]
df
dxdx = f (b)− f (a)
= f (b)i · i + f (a)i · −i
= F (b) · n+ F (a) · n= total outward flux of F across the boundary
Fundamental Theorem of Calculus
ˆ
[a,b]
df
dxdx = f (b)− f (a)
Let F = f (x)i
ˆ
[a,b]
df
dxdx = f (b)− f (a)
= f (b)i · i + f (a)i · −i
= F (b) · n+ F (a) · n= total outward flux of F across the boundary
=
ˆ
[a,b]∇ · Fdx
Integral of the differential operator acting on a field over aregion equal the sum of (or integral of ) field componentsappropriate to the operator on the boundary of the region
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