Recap of Vector Calculus - Arraytool...2015/05/01 · Recap of Vector Calculus S. R. Zinka...

Vector Algebra Vector Calculus VC - Differential Elements VC - Differential Operators Divergence & Stokes’ Theorems Summary

Recap of Vector Calculus

S. R. [email protected]

Department of Electrical & Electronics EngineeringBITS Pilani, Hyderbad Campus

May 7, 2015

Vector Calculus EE208, School of Electronics Engineering, VIT

mailto:[email protected]


Outline

1 Vector Algebra

2 Vector Calculus

3 VC - Differential Elements

4 VC - Differential Operators

5 Divergence & Stokes’ Theorems

6 Summary



Norm (Absolute/Modulus/Magnitude)

Definition

Given a vector space V over a subfield F of the complex numbers, a norm 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)



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 + x22 + . . . + 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)



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)



Norm - The Concept of Unit Circle

x 1x 2 x ∞



Addition & Subtraction

b

b

aaa+

b

Addition

b

b

a-ba-ba

Subtraction



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)



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.



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)



Cross or Vector Product - Physical Interpretation



Cross or Vector Product - Why the Name CrossProduct?

~a×~b =

∣∣∣∣∣∣x̂ ŷ ẑax ay azbx by bz

∣∣∣∣∣∣



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

∣∣∣∣∣∣



Scalar Triple Product - Physical Interpretation

base

a

bcθ

hα

Corollary

If the scalar triple product is equal to zero, then the three vectors~a,~b, and~c are coplanar, since theparallelepiped defined by them would be flat and have no volume.



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)



Vectors - Independency & Orthogonality



Outline

1 Vector Algebra

2 Vector Calculus




6 Summary



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



Typical 2D Coordinate Systems

Cartesian Polar

x = ρ cos φ

y = ρ sin φρ =

√x2 + y2

φ = tan−1( y

x

)



2D Coordinate Transformations

[AρAφ

]=

[cos φ sin φ− sin φ cos φ

] [AxAy

][

AxAy

]=

[cos φ − sin φsin φ cos φ

] [AρAφ

]



Typical 3D Coordinate Systems (RHS)

X

Y

Z

Oxy

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



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

)



Cross Product of Standard Basis Vectors

O

ρφ

z

(ρ,φ,z)

X

Y

Z

x̂× ŷ = ẑŷ× ẑ = x̂ẑ× x̂ = ŷx̂× x̂ = 0̂

ρ̂× φ̂ = ẑφ̂× ẑ = ρ̂ẑ× ρ̂ = φ̂ρ̂× ρ̂ = 0̂

and so on ...

r̂× θ̂ = φ̂θ̂ × φ̂ = r̂φ̂× r̂ = θ̂r̂× r̂ = 0̂



Dot Product of Standard Basis Vectors

O

ρφ

z

(ρ,φ,z)

X

Y

Z

x̂ · x̂ = ŷ · ŷ = ẑ · ẑ = 1x̂ · ŷ = ŷ · ẑ = x̂ · ẑ = 0

ρ̂ · ρ̂ = φ̂ · φ̂ = ẑ · ẑ = 1ρ̂ · φ̂ = φ̂ · ẑ = ẑ · ρ̂ = 0

r̂ · r̂ = θ̂ · θ̂ = φ̂ · φ̂ = 1r̂ · θ̂ = θ̂ · φ̂ = φ̂ · r̂ = 0

and so on ...



3D Coordinate TransformationsCartesian⇐⇒ Cylindrical

O

ρφ

z

(ρ,φ,z)

X

Y

Z

AρAφAz

= cos φ sin φ 0− sin φ cos φ 0

0 0 1

AxAyAz

AxAy

Az

= cos φ − sin φ 0sin φ cos φ 0

0 0 1

AρAφAz



3D Coordinate TransformationsCartesian⇐⇒ Spherical

ArAθAφ

= 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

AxAy

Az

= sin θ cos φ cos θ cos φ − sin φsin θ sin φ cos θ sin φ cos φ

cos θ − sin θ 0

ArAθAφ



3D Coordinate TransformationsCylindrical⇐⇒ Spherical

O

ρφ

z

(ρ,φ,z)

X

Y

Z

ArAθAφ

= sin θ 0 cos θcos θ 0 − sin θ

0 1 0

AρAφAz

AρAφ

Az

= sin θ cos θ 00 0 1

cos θ − sin θ 0

ArAθAφ



Would you like to see a few more coordinate systems?



Parabolic Coordinate System



Curvilinear Coordinate System

e1

e2

b1

b2

b1

b2



Outline

1 Vector Algebra

2 Vector Calculus




6 Summary



Infinitesimal Differential Elements - Cartesian - ~dl

~dl = dxx̂ + dyŷ + dzẑ



Infinitesimal Differential Elements - Cartesian - ~ds

~ds = ±dxdyẑ (or) ± dydzx̂ (or) ± dzdxŷ



Infinitesimal Differential Elements - Cartesian - dv

dv = dxdydz



Infinitesimal Differential Elements - Cylindrical - ~dl

~dl = dρρ̂ + ρdφφ̂ + dzẑ



Infinitesimal Differential Elements - Cylindrical - ~ds

~ds = ±ρdφdρẑ



Infinitesimal Differential Elements - Cylindrical - ~ds

~ds = ±ρdφdzρ̂



Infinitesimal Differential Elements - Cylindrical - dv

dv = ρdρdφdz



Infinitesimal Differential Elements - Spherical - ~dl

~dl = drr̂ + rdθθ̂ + r sin θdφφ̂



Infinitesimal Differential Elements - Spherical - ~ds

~ds = ±r2 sin θdθdφr̂



Infinitesimal Differential Elements - Spherical - dv

dv = r2 sin θdrdθdφ



Outline

1 Vector Algebra

2 Vector Calculus




6 Summary



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 · n̂|V| ds = limV→{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

)



Divergence - Physical Interpretation

V

Sn

nn

n

∇ ·~F = ∂Fx∂x

+∂Fy∂y

+∂Fz∂z



Curl

Definition

If n̂ is any unit vector, the curl of ~F is defined to be the limiting value of a closed line integral ina plane orthogonal to n̂ 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| n̂. (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]



Curl - Physical Interpretation

∇×~F =(

∂Fz∂y−

∂Fy∂z

)x̂ +

(∂Fx∂z− ∂Fz

∂x

)ŷ +

(∂Fy∂x− ∂Fx

∂y

)ẑ



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

x̂ +∂w∂y

ŷ +∂w∂z

ẑ. (13)

Properties

• ∇ (k1v + k2w) = k1∇v + k2∇w (Linearity)• ∇ (vw) = v∇w + w∇v (Product Rule)



Gradient - Physical Interpretation

∇w = ∂w∂x

x̂ +∂w∂y

ŷ +∂w∂z

ẑ



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)



Curvilinear Coordinate Systems - Divergence, Curl,and Gradient

∇ ·~v = 1h1h2h3

[∂

∂q1(h2h3v1) +

∂

∂q2(h3h1v2) +

∂

∂q3(h1h2v3)

]

∇×~v = 1h1h2h3

∣∣∣∣∣∣h1 q̂1 h1 q̂2 h1 q̂3

∂∂q1

∂∂q2

∂∂q3

h1v1 h2v2 h3v3

∣∣∣∣∣∣∇w = ∑

i

(q̂i

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).



Second Order Derivatives - DCG Chart

∇2w = 4w = ∇ · (∇w)

∇×∇×~A = ∇(∇ ·~A

)−∇2~A



Scalar Laplacian - Curvilinear Coordinate System

∇2w = 1h1h2h3

[∂

∂q1

(h2h3h1

∂w∂q1

)+

∂

∂q2

(h3h1h2

∂w∂q2

)+

∂

∂q3

(h1h2h3

∂w∂q3

)]



Outline

1 Vector Algebra

2 Vector Calculus




6 Summary



Open and Closed Surfaces

‚&˝ ˜

&¸



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 · n̂

)ds =

‹S~F · ~ds. (16)



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



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)



Stokes’ Theorem - Physical Interpretation

˛1=(∇×~F

)1· ~ds1

˛2=(∇×~F

)2· ~ds2

Sum : ∑i

˛i= ∑

i

(∇×~F

)i· ~dsi



Outline

1 Vector Algebra

2 Vector Calculus




6 Summary



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 =∣∣∣∣∣∣

x̂ ŷ ẑAx 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 ***



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

)



Coordinate Transformations (Vector)

AρAφAz

= cos φ sin φ 0− sin φ cos φ 0

0 0 1

AxAyAz

AxAy

Az

= cos φ − sin φ 0sin φ cos φ 0

0 0 1

AρAφAz

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

AxAy

Az

= sin θ cos φ cos θ cos φ − sin φsin θ sin φ cos θ sin φ cos φ

cos θ − sin θ 0

ArAθAφ

ArAθ

Aφ

= sin θ 0 cos θcos θ 0 − sin θ

0 1 0

AρAφAz

AρAφ

Az

= sin θ cos θ 00 0 1

cos θ − sin θ 0

ArAθAφ



Differential Elements

Cartesian Coordinate System:

~dl = dxx̂ + dyŷ + dzẑ

~ds = ±dxdyẑ (or) ± dydzx̂ (or) ± dzdxŷdv = dxdydz

Cylindrical Coordinate System:

~dl = dρρ̂ + ρdφφ̂ + dzẑ

~ds = ±ρdφdρẑ (or) ± ρdφdzρ̂dv = ρdρdφdz

Spherical Coordinate System:

~dl = drr̂ + rdθθ̂ + r sin θdφφ̂

~ds = ±r2 sin θdθdφr̂dv = r2 sin θdrdθdφ



Divergence, Curl, and Gradient

∇ ·~v = 1h1h2h3

[∂

∂q1(h2h3v1) +

∂

∂q2(h3h1v2) +

∂

∂q3(h1h2v3)

]

∇×~v = 1h1h2h3

∣∣∣∣∣∣h1 q̂1 h2 q̂2 h3 q̂3

∂∂q1

∂∂q2

∂∂q3

h1v1 h2v2 h3v3

∣∣∣∣∣∣∇w = ∑

i

(q̂i

1hi

∂w∂qi

)

∇2w = 1h1h2h3

[∂

∂q1

(h2h3h1

∂w∂q1

)+

∂

∂q2

(h3h1h2

∂w∂q2

)+

∂

∂q3

(h1h2h3

∂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)



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) ***



Important Integral Identities

• ˝V(∇ ·~F

)dv =

‚S~F · ~ds (Divergence Theorem)

• ˜S(∇×~F

)· ~ds =

¸C~F · ~dl (Stokes’ Theorem)


Date post:	29-Jan-2021
Category:	Documents
Upload:	others
View:	5 times
Download:	0 times

Recap of Vector Calculus - Arraytool...2015/05/01 · Recap of Vector Calculus S. R. Zinka...

Documents