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Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Vector Calculus
S. R. [email protected]
School of Electronics EngineeringVellore Institute of Technology
July 16, 2013
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Outline
1 Vectors
2 Coordinate Systems
3 VC - Differential Elements
4 VC - Differential Operators
5 Important Theorems
6 Summary
7 Problems
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Outline
1 Vectors
2 Coordinate Systems
3 VC - Differential Elements
4 VC - Differential Operators
5 Important Theorems
6 Summary
7 Problems
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Norm (Absolute/Modulus/Magnitude) ∗ ∗ ∗
Definition
Given a vector space V over a subfield F of the complex numbers, a norma on V is a function‖‖ : V → R with the following properties:For all a ∈ F and all~u,~v ∈ V,
1 ‖a~v‖ = |a| ‖~v‖ (positive scalability).
2 ‖~u +~v‖ ≤ ‖~u‖+ ‖~v‖ (triangular inequality)
3 If ‖~v‖ = 0 then~v is the zero vector~0 (separates points)
aSometimes the vertical line, Unicode Ux007c (|), is used (e.g., |v|), but this latter notation isgenerally discouraged, because it is also used to denote the absolute value of scalars and thedeterminant of matrices.
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Norm - A Few Examples ∗ ∗ ∗
Euclidean Norm
• On an n-dimensional Euclidean space Rn, the intuitive notion of length of the vectorx = (x1, x2, . . . , xn) is captured by the formula
‖x‖2 :=√
x21 + x2
2 + . . . + x2n. (1)
• On an n-dimensional complex space Cn, the most common norm is
‖z‖2 :=√|z1|2 + |z2|2 + . . . + |zn|2. (2)
Taxicab Norm / Manhattan Norm
• The name relates to the distance a taxi has to drive in a rectangular street grid to get from theorigin to the point x. It is defined as
‖x‖1 :=n
∑i=1|xi| . (3)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Norm - A Few Examples ∗ ∗ ∗
Maximum Norm
• Maximum norm is defined as
‖x‖∞ := max (|x1| , |x2| , . . . , |xn|) . (4)
p-norm
• p-norm is defined as
‖x‖p :=
(n
∑i=1|xi|p
)1/p
. (5)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Norm - The Concept of Unit Circle ∗ ∗ ∗
x 1x 2 x ∞
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Addition & Subtraction
b
b
aaa+
b
Addition
b
b
a-ba-ba
Subtraction
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Dot or Scalar Product
Definition
The dot product of two vectors,~a = [a1, a2, . . . , an] and~b = [b1, b2, . . . , bn] in a vector space of dimen-sion n is defined as
~a ·~b =n
∑i=1
aibi = a1b1 + a2b2 + . . . + anbn = ‖~a‖∥∥∥~b∥∥∥ cos θ. (6)
Properties
• ~a ·~b =~b ·~a (commutative)
• ~a ·(~b +~c
)=~a ·~b +~a ·~c (distributive over vector addition)
• ~a ·(
r~b +~c)= r
(~a ·~b
)+~a ·~c (bilinear)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Dot or Scalar Product - Physical Interpretation
Projection of~a in the direction of~b, ab is given by
ab =~a ·~b∥∥∥~b∥∥∥ (7)
Corollary
If~a ·~b =~a ·~c and~a 6= ~0, then we can write: ~a ·(~b−~c
)= 0 by the distributive law; the result above
says this just means that~a is perpendicular to(~b−~c
), which still allows
(~b−~c
)6=~0, and therefore
~b 6=~c.
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Cross or Vector Product
Definition
The cross product~a×~b is defined as a vector~c that is perpendicular to both~a and~b, with a directiongiven by the right-hand rule and a magnitude equal to the area of the parallelogram that the vectorsspan.
~a×~b =(‖~a‖
∥∥∥~b∥∥∥ sin θ)~n (8)
Properties
• ~a×~b = −~b×~a (anti-commutative)
• ~a×(~b +~c
)=~a×~b +~a×~c (distributive over vector addition)
• ~a×(
r~b +~c)= r
(~a×~b
)+~a×~c (bilinear)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Cross or Vector Product - Physical Interpretation
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Cross or Vector Product - Why the Name CrossProduct?
~a×~b =
∣∣∣∣∣∣x y zax ay azbx by bz
∣∣∣∣∣∣
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Scalar Triple Product
Definition
The scalar triple product of three vectors is defined as the dot product of one of the vectors with thecross product of the other two,
~a ·(~b×~c
)=~b · (~c×~a) =~c ·
(~a×~b
). (9)
Properties
• ~a ·(~b×~c
)= −~a ·
(~c×~b
)• ~a ·
(~b×~c
)=
∣∣∣∣∣∣a1 a2 a3b1 b2 b3c1 c2 c3
∣∣∣∣∣∣
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Scalar Triple Product - Physical Interpretation
base
a
b
c
θ
hα
Corollary
If the scalar triple product is equal to zero, then the three vectors~a,~b, and~c are coplanar, since the
parallelepiped defined by them would be flat and have no volume.
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Vector Triple Product
Definition
The vector triple product is defined as the cross product of one vector with the cross product of theother two,
~a×(~b×~c
)=~b (~a ·~c)−~c
(~a ·~b
). (10)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Vectors - Independency & Orthogonality
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Outline
1 Vectors
2 Coordinate Systems
3 VC - Differential Elements
4 VC - Differential Operators
5 Important Theorems
6 Summary
7 Problems
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Remember Complex Numbers?
Cartesian Polar
Euler’s formula is our jewel and one of the most remarkable, almost astounding, formulas in all
of mathematics - Richard Feynman
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Typical 2D Coordinate Systems
Cartesian Polar
x = ρ cos φ
y = ρ sin φ
ρ =√
x2 + y2
φ = tan−1( y
x
)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
2D Coordinate Transformations
[Aρ
Aφ
]=
[cos φ sin φ− sin φ cos φ
] [AxAy
][
AxAy
]=
[cos φ − sin φsin φ cos φ
] [Aρ
Aφ
]
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Typical 3D Coordinate Systems (RHS)
X
Y
Z
O
xy
z
(x,y,z)
Cartesian
O
ρφ
z
(ρ,φ,z)
X
Y
Z
Cylendrical
x = ρ cos φ
y = ρ sin φ
z = z
ρ =√
x2 + y2
φ = tan−1( y
x
)z = z
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Typical 3D Coordinate Systems (RHS)
Spherical
x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
r =√
x2 + y2 + z2
θ = cos−1
(z√
x2 + y2 + z2
)
φ = tan−1( y
x
)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Cross Product of Standard Basis Vectors
O
ρφ
z
(ρ,φ,z)
X
Y
Z
x× y = z
y× z = x
z× x = y
x× x = 0
ρ× φ = z
φ× z = ρ
z× ρ = φ
ρ× ρ = 0
and so on ...
r× θ = φ
θ × φ = r
φ× r = θ
r× r = 0
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Dot Product of Standard Basis Vectors
O
ρφ
z
(ρ,φ,z)
X
Y
Z
x · x = y · y = z · z = 1
x · y = y · z = x · z = 0
ρ · ρ = φ · φ = z · z = 1
ρ · φ = φ · z = z · ρ = 0
r · r = θ · θ = φ · φ = 1
r · θ = θ · φ = φ · r = 0
and so on ...
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
3D Coordinate TransformationsCartesian⇐⇒ Cylindrical
O
ρφ
z
(ρ,φ,z)
X
Y
Z
Aρ
Aφ
Az
=
cos φ sin φ 0− sin φ cos φ 0
0 0 1
AxAyAz
Ax
AyAz
=
cos φ − sin φ 0sin φ cos φ 0
0 0 1
Aρ
Aφ
Az
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
3D Coordinate TransformationsCartesian⇐⇒ Spherical
ArAθ
Aφ
=
sin θ cos φ sin θ sin φ cos θcos θ cos φ cos θ sin φ − sin θ− sin φ cos φ 0
AxAyAz
Ax
AyAz
=
sin θ cos φ cos θ cos φ − sin φsin θ sin φ cos θ sin φ cos φ
cos θ − sin θ 0
ArAθ
Aφ
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
3D Coordinate TransformationsCylindrical⇐⇒ Spherical
O
ρφ
z
(ρ,φ,z)
X
Y
Z
ArAθ
Aφ
=
sin θ 0 cos θcos θ 0 − sin θ
0 1 0
Aρ
Aφ
Az
Aρ
Aφ
Az
=
sin θ cos θ 00 0 1
cos θ − sin θ 0
ArAθ
Aφ
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Would you like to see a few more coordinate systems?
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Parabolic Coordinate System
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Curvilinear Coordinate System
e1
e2
b1
b2
b1
b2
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Outline
1 Vectors
2 Coordinate Systems
3 VC - Differential Elements
4 VC - Differential Operators
5 Important Theorems
6 Summary
7 Problems
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Cartesian - ~dl
~dl = dxx + dyy + dzz
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Cartesian - ~ds
~ds = ±dxdyz (or) ± dydzx (or) ± dzdxy
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Cartesian - dv
dv = dxdydz
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Cylindrical - ~dl
~dl = dρρ + ρdφφ + dzz
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Cylindrical - ~ds
~ds = ±ρdφdρz
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Cylindrical - ~ds
~ds = ±ρdφdzρ
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Cylindrical - dv
dv = ρdρdφdz
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Spherical - ~dl
~dl = drr + rdθθ + r sin θdφφ
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Spherical - ~ds
~ds = ±r2 sin θdθdφr
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Infinitesimal Differential Elements - Spherical - dv
dv = r2 sin θdrdθdφ
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Outline
1 Vectors
2 Coordinate Systems
3 VC - Differential Elements
4 VC - Differential Operators
5 Important Theorems
6 Summary
7 Problems
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Divergence
Definition
The divergence of a vector field ~F at a point P is defined as the limit of the net flow of ~F across thesmooth boundary of a three dimensional region V divided by the volume of V as V shrinks to P.Formally,
div(~F (P)
)= ∇ ·~F = lim
V→{P}
‹S(V)
~F · n|V| ds = lim
V→{P}
‹S(V)
~F · ~ds|V| . (11)
Properties
• ∇ ·(
k1~A + k2~B)= k1∇ ·~A + k2∇ ·~B (linearity)
• ∇ ·(
w~A)= w∇ ·~A +~A · ∇w
• ∇ ·(~A×~B
)= ~B ·
(∇×~A
)−~A ·
(∇×~B
)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Divergence - Physical Interpretation
V
Sn
nn
n
∇ ·~F =∂Fx
∂x+
∂Fy
∂y+
∂Fz
∂z
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Curl
Definition
If n is any unit vector, the curl of ~F is defined to be the limiting value of a closed line integral ina plane orthogonal to n as the path used in the integral becomes infinitesimally close to the point,divided by the area enclosed.
curl(~F (P)
)= ∇×~F = lim
A→0
˛C
~F · ~dl|A| n. (12)
Properties
• ∇×(
k1~A + k2~B)= k1∇×~A + k2∇×~B (linearity)
• ∇×(
w~A)= w∇×~A−~A×∇w
• ∇×(~A×~B
)=[~A(∇ ·~B
)−~B
(∇ ·~A
)]−[(
~A · ∇)~B−
(~B · ∇
)~A]
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Curl - Physical Interpretation
∇×~F =
(∂Fz
∂y−
∂Fy
∂z
)x +
(∂Fx
∂z− ∂Fz
∂x
)y +
(∂Fy
∂x− ∂Fx
∂y
)z
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Gradient
Definition
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of thegreatest rate of increase of the scalar field, and whose magnitude is that rate of increase,
grad (w) = ∇w =∂w∂x
x +∂w∂y
y +∂w∂z
z. (13)
Properties
• ∇ (k1v + k2w) = k1∇v + k2∇w (Linearity)
• ∇ (vw) = v∇w + w∇v (Product Rule)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Gradient - Physical Interpretation
∇w =∂w∂x
x +∂w∂y
y +∂w∂z
z
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Solenoidal and Lamellar Fields
Definition
In vector calculus a solenoidal vector field (also known as an incompressible vector field) is a vectorfield~v with divergence zero at all points in the field:
∇ ·~v = 0. (14)
Definition
A vector field is said to be lamellar or irrotational if its curl is zero. That is, if
∇×~v =~0. (15)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Curvilinear Coordinate Systems - Divergence, Curl,and Gradient
∇ ·~v =1
h1h2h3
[∂
∂q1(h2h3v1) +
∂
∂q2(h3h1v2) +
∂
∂q3(h1h2v3)
]
∇×~v =1
h1h2h3
∣∣∣∣∣∣h1 q1 h1 q2 h1 q3
∂∂q1
∂∂q2
∂∂q3
h1v1 h2v2 h3v3
∣∣∣∣∣∣∇w = ∑
i
(qi
1hi
∂w∂qi
)
where
• when (q1, q2, q3) = (x, y, z) =⇒ (h1, h2, h3) = (1, 1, 1),• when (q1, q2, q3) = (ρ, φ, z) =⇒ (h1, h2, h3) = (1, ρ, 1), and
• when (q1, q2, q3) = (r, θ, φ) =⇒ (h1, h2, h3) = (1, r, r sin θ).
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Second Order Derivatives - DCG Chart
∇2w = 4w = ∇ · (∇w)
∇×∇×~A = ∇(∇ ·~A
)−∇2~A
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Scalar Laplacian - Curvilinear Coordinate System
∇2w =1
h1h2h3
[∂
∂q1
(h2h3
h1
∂w∂q1
)+
∂
∂q2
(h3h1
h2
∂w∂q2
)+
∂
∂q3
(h1h2
h3
∂w∂q3
)]
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Outline
1 Vectors
2 Coordinate Systems
3 VC - Differential Elements
4 VC - Differential Operators
5 Important Theorems
6 Summary
7 Problems
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Open and Closed Surfaces
‚&˝ ˜
&¸
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Divergence Theorem
Definition
Suppose V is a subset of Rn (in the case of n = 3, V represents a volume in 3D space) which is compactand has a piecewise smooth boundary S. If~F is a continuously differentiable vector field defined ona neighborhood of V, then we have
˚V
(∇ ·~F
)dv =
‹S
(~F · n
)ds =
‹S~F · ~ds. (16)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Divergence Theorem - Physical Interpretation
[F (y + ∆y)− F (y)]∆x∆z =(∇ ·~F
)vol1× vol1
[F (y + 2∆y)− F (y + ∆y)]∆x∆z =(∇ ·~F
)vol2× vol2
Sum : [F (y + 2∆y)− F (y)]∆x∆z = ∑i
(∇ ·~F
)voli× voli
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Stokes’ Theorem
Definition
The surface integral of the curl of a vector field over a surface S in Euclidean three-space is relatedto the the line integral of the vector field over its boundary as
¨S
(∇×~F
)· ~ds =
˛C~F · ~dl. (17)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Stokes’ Theorem - Physical Interpretation
˛1=(∇×~F
)1· ~ds1
˛2=(∇×~F
)2· ~ds2
Sum : ∑i
˛i= ∑
i
(∇×~F
)i· ~dsi
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Outline
1 Vectors
2 Coordinate Systems
3 VC - Differential Elements
4 VC - Differential Operators
5 Important Theorems
6 Summary
7 Problems
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Important Vectorial Identities
• A · B = B ·A = ‖A‖ ‖B‖ cos θ
• AB = A·B‖B‖
B‖B‖
• A× B = −B×A = (‖A‖ ‖B‖ sin θ)~n =
∣∣∣∣∣∣x y z
Ax Ay AzBx By Bz
∣∣∣∣∣∣• A · (B×C) = B · (C×A) = C · (A× B) =
∣∣∣∣∣∣Ax Ay AzBx By BzCx Cy Cz
∣∣∣∣∣∣• A× (B×C) = B (A ·C)−C (A · B)• (A× B) · (C×D) = (A ·C) (B ·D)− (B ·C) (A ·D) ***
• (A× B)× (C×D) = (A · B×D)C− (A · B×C)D ***
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Coordinate Transformations (Point)
x = ρ cos φ
y = ρ sin φ
ρ =√
x2 + y2
φ = tan−1( y
x
)x = r sin θ cos φ
y = r sin θ sin φ
z = r cos θ
r =√
x2 + y2 + z2
θ = cos−1
(z√
x2 + y2 + z2
)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Coordinate Transformations (Vector)
Aρ
Aφ
Az
=
cos φ sin φ 0− sin φ cos φ 0
0 0 1
AxAyAz
Ax
AyAz
=
cos φ − sin φ 0sin φ cos φ 0
0 0 1
Aρ
Aφ
Az
Ar
Aθ
Aφ
=
sin θ cos φ sin θ sin φ cos θcos θ cos φ cos θ sin φ − sin θ− sin φ cos φ 0
AxAyAz
Ax
AyAz
=
sin θ cos φ cos θ cos φ − sin φsin θ sin φ cos θ sin φ cos φ
cos θ − sin θ 0
ArAθ
Aφ
Ar
Aθ
Aφ
=
sin θ 0 cos θcos θ 0 − sin θ
0 1 0
Aρ
Aφ
Az
Aρ
Aφ
Az
=
sin θ cos θ 00 0 1
cos θ − sin θ 0
ArAθ
Aφ
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Differential Elements
Cartesian Coordinate System:
~dl = dxx + dyy + dzz
~ds = ±dxdyz (or) ± dydzx (or) ± dzdxy
dv = dxdydz
Cylindrical Coordinate System:
~dl = dρρ + ρdφφ + dzz
~ds = ±ρdφdρz (or) ± ρdφdzρ
dv = ρdρdφdz
Spherical Coordinate System:
~dl = drr + rdθθ + r sin θdφφ
~ds = ±r2 sin θdθdφr
dv = r2 sin θdrdθdφ
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Divergence, Curl, and Gradient
∇ ·~v =1
h1h2h3
[∂
∂q1(h2h3v1) +
∂
∂q2(h3h1v2) +
∂
∂q3(h1h2v3)
]
∇×~v =1
h1h2h3
∣∣∣∣∣∣h1 q1 h2 q2 h3 q3
∂∂q1
∂∂q2
∂∂q3
h1v1 h2v2 h3v3
∣∣∣∣∣∣∇w = ∑
i
(qi
1hi
∂w∂qi
)
∇2w =1
h1h2h3
[∂
∂q1
(h2h3
h1
∂w∂q1
)+
∂
∂q2
(h3h1
h2
∂w∂q2
)+
∂
∂q3
(h1h2
h3
∂w∂q3
)]
where,
(q1, q2, q3) (v1, v2, v3) (h1, h2, h3)
Catersian (x, y, z)(vx, vy, vz
)(1, 1, 1)
Cylindrical (ρ, φ, z)(vρ , vφ , vz
)(1, ρ, 1)
Spherical (r, θ, φ)(vr, vθ , vφ
)(1, r, r sin θ)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Important Differential Identities
• ∇ (vw) = v∇w + w∇v• ∇ (A · B) =
(A · ∇)B + (B · ∇)A + A× (∇× B) + B× (∇×A)***
• ∇ · (wA) = w∇ ·A + A · ∇w• ∇ · (A× B) = B · (∇×A)−A · (∇× B)• ∇× (wA) = w∇×A−A×∇w ***• ∇× (A× B) =
[A (∇ · B)− B (∇ ·A)]− [(A · ∇)B− (B · ∇)A] ***• ∇×∇×A = ∇ (∇ ·A)−∇2A• ∇ |r| = r
|r| ***
• ∇ 1|r| = −
r|r|3
***
• ∇.(
r|r|3
)= −∇2
(1|r|
)= 4πδ (r) ***
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Important Integral Identities
• ˝V
(∇ ·~F
)dv =
‚S~F · ~ds (Divergence Theorem)
• ˜S
(∇×~F
)· ~ds =
¸C~F · ~dl (Stokes’ Theorem)
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Outline
1 Vectors
2 Coordinate Systems
3 VC - Differential Elements
4 VC - Differential Operators
5 Important Theorems
6 Summary
7 Problems
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Divergence
1 Evaluate the ∇ ·~G at
• PCart(2,−3, 4), if ~G = xx + y2y + z3 z;• PCyl(2, 110◦ ,−1), if ~G = 2ρz2 sin2 φρ + ρz2 sin 2φφ + 2ρ2z sin2 φz; and• PSpherical(1.5, 30◦ , 50◦), if ~G = 2r sin θ cos φr + r cos θ cos φθ − r sin φφ.
Ans: 43; 9.06; 1.28 [H1, D3.7, P73]
2 Given the electric flux density, ~D = 0.3r2 r nC/m2 in free space:
• find~E(=
~Dε , where ε≈ 8.8542× 10−12F/m
)at point P (r = 2, θ = 25◦ , φ = 90◦);
• find the total charge(
ρv = ∇ · ~D)
within the sphere ‖~r‖ = 3;• find the total electric flux leaving the sphere ‖~r‖ = 4.
Ans: 135.5r V/m; 305 nC; 965 nC [H1, D3.3, P61]
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Curl
1 Calculate the value of the vector current density(~Je = ∇× ~H
):
• in Cartesian coordinates at PCart(2, 3, 4), if ~H = x2zy− y2xz;• in cylindrical coordinates at PCyl(1.5, 90◦ , 0.5), if ~H = 2
ρ (cos 0.2φ) ρ; and
• in spherical coordinates at PSpherical(2, 30◦ , 20◦), if ~H = 1sin θ θ.
Ans: −16x + 9y + 16z A/m2; 0.0549z A/m2; φ A/m2 [H1, D8.7, P246]
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Gradient
1 Given the potential field, V = 2x2y− 5z, find the electric field intensity (~E = −∇V) at a givenpoint PCart (x, y, z).Ans: −4xyx− 2x2y + 5z V/m [H1, E4.3, P104]
2 Given the potential field, V = 100z2+1
ρ cos φ V, find the electric field intensity (~E = −∇V) at a
given point PCyl (3, 60◦ , 2).
Ans: −10ρ + 17.32φ + 24z V/m [H1, D4.8, P106]
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Divergence Theorem
1 Evaluate both sides of the divergence theorem for the field , ~D = 2xyx + x2yand therectangular parallelepiped formed by the planes x = 0 and 1, y = 0 and 2, and z = 0 and 3.Ans: 12 [H1, E3.5, P77]
2 Given the field, ~D = 6ρ sin φ2 ρ + 1.5ρ cos φ
2 φ, evaluate both sides of the divergence theoremfor the region bounded by ρ = 2, φ = 0, φ = π, z = 0, and z = 5.Ans: 225; 225 [H1, D3.9, P78]
˚V
(∇ ·~F
)dv =
‹S~F · ~ds
Vector Calculus EE533, School of Electronics Engineering, VIT
Vectors Coordinate Systems VC - Differential Elements VC - Differential Operators Important Theorems Summary Problems
Stokes’ Theorem
1 Evaluate both sides of the Stokes’ theorem for the field , ~H = 6r sin φr + 18r sin θ cos φφ, andthe patch around the region, r = 4, 0 ≤ θ ≤ 0.1π, and 0 ≤ φ ≤ 0.3π.Ans: 22.2 [H1, E8.3, P248]
2 Evaluate both sides of the Stokes’ theorem for the field , ~H = 6xyx− 3y2y, and the patharound the region, 2 ≤ x ≤ 5, −1 ≤ y ≤ 1, and z = 0. Let the positive direction of ~ds be z.Ans: -126 [H1, D8.6, P251]
¨S
(∇×~F
)· ~ds =
˛C~F · ~dl
Vector Calculus EE533, School of Electronics Engineering, VIT