Chap. 8. Vector Differential Calculus. Grad. Div. Curl

Chap. 8. Vector Differential Calculus. Grad. Div. Curl

8.1. Vector Algebra in 2-Space and 3-Space

- Scalar: a quantity only with its magnitude; temperature, speed, mass, volume, … - Vector: a quantity with its magnitude and its direction; velocity, acceleration, force, …

(arrow & directed line segment) Norm of a: length of a vector a. IaI

=1: unit vector

Equality of a Vectors: a=b: same length and direction.

Components of a Vector: P(x₁,y₁,z₁)  Q(x₂,y₂,z₂) in Cartesian coordinates.

Length in Terms of Components:

Position Vector: from origin (0,0,0)  point A (x,y,z): r=[x,y,z]

Initial and termination point

[

^{x x ,y y ,z z}

]

^{[a ,a ,a ]}

PQ

a = = ₂ − _{1 2} − _{1 2} − ₁ = _{1 2 3}

2 3 2

2 2

1 a a

a

a = + +

Vector Addition, Scalar Multiplication (1) Addition:

a + b = b + a (u + v) + w = u + (v + w) a + 0 = 0 + a = a a + (-a) = 0

(2) Multiplication:

c(a + b) = ca + cb (c + k) a = ca + ka

c(ka) = cka 1a = a 0a = 0 (-1)a = -a

Unit Vectors: i, j, k

i = [1,0,0], j=[0,1,0], k=[0,0,1]

8.2. Inner Product (Dot Product)

Definition:

[ ^a

¹

^b

¹

^{, a}

²

^b

²

^{, a}

³

^b

³

]

b

a + = + + +

a a+b b

[ ^{c a}

¹

^{, c a}

²

^{, c a}

³

]

a c =

[ ^{a , a , a} ] ^{a i a j a k}

a =

_{1 2 3}

=

₁

+

₂

+

₃

0 cos

; 0 b or 0 a if 0

b a

0 b , 0 a if cos

b a b a

=

=

=

=

⋅

≠

≠

=

⋅

γ γ



=

= +

+

=

⋅

³

1 i

i i 3

3 2

2 1

1

b a b a b a b

a b a

0 b

a ⋅ =

(a is orthogonal to b; a, b=orthogonal vectors)

Theorem 1:

The inner product of two nonzero vectors is zero iff these vectors are perpendicular. Length and Angle in Terms of Inner Product:

Ex. 1) a=[1,2,0], b=[3,-2,1], angle btw a and b ? General Properties of Inner Products:

b b a a

b a b

a b cos a

: vectors two

btw angle

a a a

: a of length

⋅

⋅

= ⋅

= ⋅

⋅

=

γ

[ ^q

¹

^{a + q}

²

^b ] ^{⋅ c = q}

¹

^{a ⋅ c + q}

²

^{b ⋅ c a ⋅ b = b ⋅ a a ⋅ a ≥ 0} [ ^{a + b} ] ^{⋅ c = a ⋅ c + b ⋅ c}

( ^{a b} ) ^{Paralle log ram equality}

2 b

a b

a

inequality Triangle

b a

b a

inequality Schwarz

b a b a

2 2

2

2

+ − = +

+

+

≤ +

≤

⋅

Derivation of

Application of Inner Products:

Ex. 2) Work

Ex. 3) Component of a force in a given direction Ex. 5) Orthogonal straight lines in the plane

Ex. 6) Normal vector to a plane

3 3 2

2 1

1

b a b a b

a b

a ⋅ = + +

3 3 2

2 1

1 3

2 3

1 2

1 1

1 ij

3 2

1 3

2 1

b a b

a b a k k b a k

i b a j i b a i i b a b a

j i for 0

j i for 1 j

i

k b j b i b b , k a j a i a a

+ +

=

⋅ +

+

⋅ +

⋅ +

⋅

=

 ⋅

≠

=

=

⋅

+ +

= +

+

=



δ

8.3. Vector Product (Cross Product)

Definition:

In components: v = [v₁,v₂,v₃]

= [a₂b₃-a₃b₂, a₃b₁-a₁b₃, a₁b₂-a₂b₁] Right-handed triple of vectors a, b, v

i, j, k form a right-handed triple in the positive directions

How to memorize above formula:

General Properties of Vector Products:

γ sin b a v : length ,

b a

v = × =

a v b

3 2

1

3 2

1

b b

b

a a

a

k j

i b a × =

alike are

indices two

any if

0

213 , 132 , 321 ijk

if 1

312 , 231 , 123 ijk

if 1 k

j i

ijk ijk ijk 3

1 k

ijk

=

=

−

=

= +

=

=

× 

=

ε ε ε ε

( ) ^{i i j i} ( ) ^{i j )}

( c ) b a ( ) c b ( a )

i j j i ( a c c a

) c b ( ) c a ( c ) b a (

) c a ( ) b a ( ) c b ( a )

b q ( a ) b a ( q b ) a q (

×

×

≠

×

×

×

×

≠

×

×

×

≠

×

×

≠

×

× +

×

=

× +

× +

×

= +

×

×

=

×

=

×





Typical Applications of Vector Products Ex. 4) Moment of a force (I)

Ex. 5) Moment of a force (II)

Ex. 6) Velocity of a rotating body

Scalar Triple Product

:

Volume of the parallelepiped with a, b, c

Theorem 1: Three vectors form a linearly independent set iff their scalar triple product is not zero.

( )

3 2

1

3 2

1

3 2

1

3 2 1 3

2 1 3

2 1

c c

c

b b

b

a a

a c

b a ) c b a (

] c , c , c [ c ], b , b , b [ b ], a , a , a [ a

=

×

⋅

=

=

=

=

b a

c bxc

β h

) b a ( c c ) b a ( ) c b ( a )

c b a ( k ) c b a k

( = ⋅ × = × ⋅ = ⋅ ×

see matrix expression

8.4. Vector and Scalar Functions and Fields. Derivatives

- Two kinds of functions

(1) Vector functions: depending on the point p in space.

 a vector field In Cartesian coordinates,

(2) Scalar functions: depending on p.  a scalar field Ex. 1) Scalar function (Euclidean distance in space)

Ex. 2, 3) Vector function (Velocity field, force field)

Vector Calculus

- Basic concepts of vector calculus

(1) Convergence: (a: limit vector)

(vector function v of a real variable t has limit u) (2) Continuity: vector function v(t) is continuous at t=t₀.

three components are continuous at t₀.

)]

p ( v ), p ( v ), p ( v [ ) p ( v

v = =

_{1 2 3}

) p ( f f =

)]

z , y , x ( v ), z , y , x ( v ), z , y , x ( v [ ) z , y , x (

v =

_{1 2 3}

a a

lim or

0 a a

lim

_n

n n

n

− = =

∞

→

∞

→

u ) t ( v lim or

0 u ) t ( v lim

0

0 t t

t

t

− = =

→

→

) t ( v ) t ( v

lim

₀

t

t ₀

=

→

k ) t ( v j ) t ( v i ) t ( v )]

t ( v ), t ( v ), t ( v [ ) t (

v =

_{1 2 3}

=

₁

+

₂

+

₃

Derivative of a vector function Definition:

Partial Derivatives of a Vector Function

differentiable functions of n variables, t₁, …, t_n.

Partial derivative of v:

t

) t ( v ) t t

( lim v )

t ( ' v

0

t

Δ

− Δ

= +

→ Δ

v(t) v(t+Δt)

V‘(t)

)]

t ( ' v ), t ( ' v ), t ( ' v [ ) t ( '

v =

_{1 2 3}

Derivative v’(t) is obtained by differentiating each component separately.

( ) ( ) ( )

( ^{u v} ) ^{' u ' v u v '} ( ^{u v w} ) ( ^{' u ' v w} ) ( ^{u v ' w} ) ( ^{u v w '} )

' v u v ' u ' v u '

u ' v ' u v '

v c ' v c

+ +

=

× +

×

=

×

⋅ +

⋅

=

⋅ +

= +

=

k v j v i v ] v , v , v [ ) t (

v =

_{1 2 3}

=

₁

+

₂

+

₃

t k t j v t t i v t t

v t

t k v

t j v t i v

t v t

v

m l

3 2

m l

2 2

m l

1 2

m l

2

l 3 l

2 l

1

l

∂ ∂

+ ∂

∂

∂ + ∂

∂

∂

= ∂

∂

∂

∂

∂ + ∂

∂ + ∂

∂

= ∂

∂

∂