Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 309 EXERCISES AND EXAMPLES with solutions 1. What conditions should satisfy vectors , ,a b c to form a triangle. Solution: + + =a b c 0 2. The set of all geometric vectors V with operations addition of vectors and

multiplication by a scalar form a vector space ( )V , ,+ ⋅ . Check if the conditions of Definition of a Linear Vector Space are satisfied (III.1, p.162; IV.1.3, p.210).

3. Prove:

a) If set of vectors includes a zero vector, then it is linearly dependent.

b) If two vectors are linearly dependent, then they are collinear.

c) Any three linearly dependent vectors are coplanar (lie in the same plane).

c) Any four vectors are linearly dependent.

4. Let the point P be defined by the position vector a . Derive a vector equation which traces points in the line which goes through the point P in the direction of vector u . Solution: t= +r a u t−∞ < < ∞ 5. Let position vectors , ,a b c define points in the same line. Show that

α β= +c a b if and only if 1α β+ = Solution: ( )s− = −c a b a s s= + −c b a a ( )s 1 s= + −c b a s 1 s 1+ − =

( )tr

a

uP

0

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 310

6. Consider a set of vectors 1 2 n, ,...,a a a and some vector u . Then ( ) ( ) ( ) ( )1 2 n 1 2 2... ...+ + + = + + +u u uua a a a a a the projection of the sum of vectors in the direction of vector u is equal to the sum of projections (enough to show for two vectors).

7. (Law of sines) a bsin A cos B

=

8. (Law of cosine) Consider three vectors , ,a b c such that + =a b c . Using the dot product, derive the law of cosine:

2 2 2c a b 2abcosγ= + − 9. Consider a vector equation c⋅ =a x 3, V∈a x c∈

Is it possible to define an “operation division by a vector” 1a

to find a solution

by

c =xa

?

10. Show that ( ) ( )⊥ ⋅ − ⋅c b a c a b c

11. Derive vector equation of the plane containing the fixed point P and orthogonal to the fixed vector u Solution: a is the vector which defines point P

( ) 0− ⋅ =a r u

0⋅ − ⋅ =a u r u

⋅ = ⋅r u a u

c⋅ =r u

1 1 2 2 3 3 1 1 2 2 3 3p x p x p x p u p u p u+ + = + +

b sin A a sin B=

bcosA

acosB

ab

A B

C

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 311

12. Find the distance form the point P to the plane c⋅ =r u .

13. Point started motion with the constant velocity v at the point P . What time is needed for the point to reach the plane c⋅ =r u . 14. What is the angle between the vectors 1 2 4= + −a i i i and 1 22 2= − +b i i i 15. Derive the equation for ( )sin α β+ by calculation of ×a b and its projection on z axis.

×a b ( ) 3sin α β= − + i

( ) x y y xz a b a b× = −a b

xa = xb =

ya = yb =

16. Let ( ) ( ) α β× × × = +a b c d a b . Find α and β . Hint: In ( ) ( ) ( )× × = ⋅ − ⋅a b c a c b b c a replace c by ×c d .

17. Derive

( ) ( )d d d ddt dt dt dt

⋅ × = ⋅ × + ⋅ × + ⋅ ×

a u va u v u v a v a u

18. Derive

( ) ( )d d d ddt dt dt dt

× × = × × + × × + × ×

a u va u v u v a v a u

19. Verify properties of Vectors: Commutative law: + = +a b b a Associative law: ( ) ( )+ + = + +a b c a b c Distributive laws: ( )+ ⋅ = ⋅ + ⋅a b c a c b c

( ) ( ) ( )× + = × + ×a b c a b a c ( ) ( ) ( )+ × = × + ×a b c a c b c Anticommutativity: × = − ×b a a b

Triple products: ( )1 2 3

1 2 3

1 2 3

a a a b b b

c c c⋅ × =a b c

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 312

20. Verify:

The modulus of the triple product ( )⋅ ×a b c is equal to the volume of the parallelepiped constructed from vectors , ,a b c .

21. Prove the identiries:

a) ( ) ( ) ( )× × = ⋅ − ⋅a b c a c b a b c

b) ( ) ( ) ( )× × = ⋅ − ⋅a b c a c b b c a

d) ( ) ( ) ( )( ) ( )( )× ⋅ × = ⋅ ⋅ − ⋅ ⋅a b c d a c b d a d b c Lagrange Identity 22. Differentiation:.

Let ( )tu and ( )tv be differentiable vector functions. Prove that:

( )c c′ ′=u u

( )′ ′ ′+ = +u v u v

( )′ ′ ′⋅ = ⋅ + ⋅u v u v u v

( )′ ′ ′× = × + ×u v u v u v

23. Divergence identities:

( )div c cdiv=a a

( ) 2div f g f g f g∇ = ∇ +∇ ⋅∇

( )div f fdiv f= + ⋅∇u u u

( )div curl curl× = ⋅ − ⋅u v v u u v

( )div curl 0=u

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 313

24. A point with the mass m is moving under the force α= −F r . Find the trajectory of the point subject to the initial conditions: ( ) 00 =r r ( ) 00′ =r v

The governing equation: m α′′ = −r r

0mα′′ + =r r

This 2nd order ODE with constant coefficients for unknown vector function can

be solved similar to ODE for scalar function. Since 0mα

> ,

( )t cos t sin tm mα α

= +r A B

where arbitrary vector coefficient can be determined from the initial conditions:

0mα

=A v

0=B r

Let vectors ( ) 00 =r r and ( ) 00′ =r v are not collinear. Let vectors ( ) 00 =r r and ( ) 00′ =r v be in the xy-plane. Sketch the trajectory of the point. Make the conclusions.

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 314

25. Equation of Continuity

Consider the fluid flow with the velocity field ( )v r . The flux of the velocity field through some surface S determines the volume of fluid flowing through the surface S per unit time:

S

q dS= ⋅∫ v n

Then for incompressible fluid

m qρ=S

dS= ⋅∫ v n

is the mass of fluid flowing through the surface S per unit time.

For compressible fluid, density varies in space ( )ρ ρ= r , so

( )S

m dSρ= ⋅∫ r v n

This equation represents the net amount of fluid flowing through the surface S with normal vector n representing the positive direction. Let now S be the closed surface of some finite control volume V containing the point of space r . In the stationary case, without any sources or sinks,

( )S

m dS 0ρ= ⋅ =∫ r v n

i.e. the mass of fluid flowing into V is equal to the mass of fluid flowing out of V. The mass of fluid in the control volume is determined by the volume integral

V

dVρ∫

In the non-stationary fluid flow, density depends on time

( ),tρ ρ= r . Then the conservation of mass in the control volume yields that the change of mass in the control volume V is equal to the mass flowing through the surface of the volume:

V S

dV dSt

ρ ρ∂ = − ⋅∂ ∫ ∫ v n

(sign minus because negative direction is toward the control volume). For fixed boundaries of control differentiation can be moved inside of the integral

V S

dV dS 0tρ ρ∂ + ⋅ =∂∫ ∫ v n

Apply the divergence theorem (Equation 84 b) to replace the surface integral by the volume integral

( )V V

dV div dv 0tρ ρ∂ + =∂∫ ∫ v

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 315

Adding integrals, we obtain

( )V

div dV 0tρ ρ∂ + = ∂ ∫

v

Because this equation is valid for any control volume containing the point r and the velocity and density fields are continuous, the integrand should be equal to zero provided that it also is continuous:

( )div 0tρ ρ∂ + =∂

v

This equation is called the equation of continuity. Using operator nabla, we can rewrite it in the form:

( ) 0tρ ρ∂ +∇ ⋅ =∂

v

For incompressible fluid, density does not depend on location,

and 0tρ∂≈

∂ is negligible. Then the equation of continuity

reduces to

0∇⋅ =v for both stationary and non-stationary flow. 26. Write equation of continuity to Cartesian, cylindrical and spherical coordinates.

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 316 27. Find the equation of the line obtained by the intersection of two planes a⋅ =r n b⋅ =r m Write the solution for ( )2, 1,3= −n , ( )1,2, 1= −m , a b 1= = . 28. Find the intersection of three planes a⋅ =r n

b⋅ =r m c⋅ =r k

Write the solution for

( )2, 1,3= −n , ( )1,2, 1= −m , ( )0, 1,2= −k and a b c 1= = = .

29. Sketch the space curve defined by the vector function: a) ( )t sin t cos t= +r i j b) ( ) 2t t t= +r i j c) ( )t sin t cos t 3= + +r i j k d) ( )t 2 sin t 3cos t 2= + +r i j k e) ( ) 2t t t t= + +r i j k 30. Find the first and the second derivatives of the vector functions: a) ( )t sin t cos t 2t= + +r i j k b) ( ) 2t t sin t t cos t t= + +r i j k c) ( ) t 2t tt e e te= + +r i j k 31. Using integration find the position vector and sketch the trajectory ( )tr of a

particle moving with acceleration ( )t 9= −a j if it had the initial position at 0 5=r j and the initial velocity 0 4 5= +v i j .

32. Find the equation of the surface of the circular cylinder of radius a and the axis going through the origin in the direction of vector u .

33. Let the particle be traveling along the line with a constant velocity v .

If at t 0= the particle was at the point ( )0 0 0 0x , y ,z=r , determine at which moment of time it crosses the sphere c⋅ =r r .

34. For vector functions ( ) ( ) ( ) 3t , t , t : →a u v derive

a) ( )′ ′ ′⋅ = ⋅ + ⋅u v u v u v

b) ( )′ ′ ′× = × + ×u v u v u v

c) ( ) ( )d d d ddt dt dt dt

⋅ × = ⋅ × + ⋅ × + ⋅ ×

a u va u v u v a v a u

d) ( ) ( )d d d ddt dt dt dt

× × = × × + × × + × ×

a u va u v u v a v a u

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 317

35. Length of the path:

Let a curve be traced by the vector function ( )tr , then

( ) ( )0

t 2

t

s t dτ τ′ = ∫ r .

a) Find the length of the curve (sketch the graph):

( )t 5 cos t ,5 sin t ,4t=r 0 t 4π≤ ≤ b) Derive the formula

( )b 2

a

s 1 f x dx′ = + ∫ for the length of the curve defined by the explicit equation

( )y f x= a x b≤ ≤

36. Compute the gradient for the scalar fields:

a) ( ) 2 2x 4xy yϕ = − +r b) ( ) ( )2 2 2ln x y zϕ = + +r c) ( ) 2

xyz

ϕ =r

37. Find the divergence and the curl of the following vector fields:

a) ( ) xy yz xz= + +a r i j k b) ( ) ( )2 yz zx y ye xye− −= − + +a r i j k c) ( ) 2xy ln z xyz y= + +a r i j k

38. Let ( ) ( ) ( ) 3 3, , : →a r u r v r be vector fields, and ( ) 3f : →r be scalar field, and c∈ is a constant. Prove the divergence identities:

a) ( )div c cdiv=a a b) ( )div f fdiv f= + ⋅∇u u u c) ( )div curl curl× = ⋅ − ⋅u v v u u v d) ( )div curl 0=u 39. Evaluate

C

d⋅∫a r for:

a) ( ) y 2x= +a r i j , ( )t t ln t= +r i j , 1 t e≤ ≤

b) ( ) xy yz xy= + +a r i j k , ( )t sin t cos t= +r i j , 0 t π≤ ≤

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 318

40. A point with the mass m is moving under the force α= −F r , 0α > . Find the trajectory of the point subject to the initial conditions:

( ) 00 =r r ( ) 00′ =r v

The governing equation (Newton’s Second Law): m α′′ = −r r

0mα′′ + =r r

This 2nd order ODE with constant coefficients for vector function ( )tr can be solved similar to an ODE for a scalar function. Since m 0α > ,

( )t cos t sin tm mα α

= +r A B

where the arbitrary vector coefficients can be determined from the initial conditions:

0=A r , 0mα

=B v

Let vectors ( ) 00 =r r and ( ) 00′ =r v not be collinear. Let vectors ( ) 00 =r r and ( ) 00′ =r v be in the xy-plane. Sketch the trajectory of the point. Make some conclusions.

41. [Zill, p.344] Use Gram-Schmidt Orthogonalization Process to convert the following set of vectors

( ) ( ) ( ){ }1,0,1 , 0,1,0 , 1,0,1− to orthonormal basis.

Use obtained basis to represent a vector ( )10,7, 13− . 42. Given the heat flux in direction through the semi-annual domain

(normal distribution of heat flux over area ):

What is the rate of heat transfer through the area ?

zA

( )q x, y′′2 2

1x y

=+ 2

Wm

q A

A

2 3x

y ( )q x, y′′ 2 21

x y=

+

x

y

z

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 319

43. a) Find surface area of a sphere with radius :

b) Let and temperature distribution is given by

Find the surface average temperature:

44. Consider the long rectangular column with the known temperature distribution

( ) ( ) 3T T x, y,z :V= ⊂ →r scalar field ( ) 2 2 225T x, y,z

x y z 4=

+ + +

L 4.0= M 3.0= K 2.0= k 2= thermal conductivity Describe and visualize the heat transfer in the column. Determine:

a) average temperature in the surface S ( ){ }S x,0,z 0 x L,0 z K= ≤ ≤ ≤ ≤

b) average temperature in the region

( ){ }V x, y,z 0 x L,0 y M ,0 z K= ≤ ≤ ≤ ≤ ≤ ≤

c) the rate of heat transfer through the surface S

45. A measured local for flow over a sphere of radius is correlated as a function of polar angle only:

, (and symmetric over the other half). What is the averaged over the sphere surface?

R

R 2=

( ) 2 2 2T x, y,z x y z 1= + + +

averT ?=

Nu R 0.05=θ

( ) 3 2Nu , 15 50 500θ φ θ θ= − + 0 θ π≤ ≤

Nu

y

S

R

x

z

x

y

z

n

KS

L

M

( )T x, y,z

0

q ?=

V

flow

( )Nu θ

R

θ

( )Nu θ

θ

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 320 46. [RF] What is meant?

a) ia

b) ja c) ijA d) ijkA

47. What are? a) ij ija b =

b) ij ika b =

c) ij mna b = d) ii δ = e) ij ij δ δ = f) ij ik jk δ δ δ = g) ij i j x xδ = 48. [DM] Show that operator i j∂ ∂ is symmetric. 49. [DM] Prove that the product ij ijS T is zero if ijS is symmetric, ij jiS S= , and ijT is antisymmetric, ij jiT T= − . 50. [RF] Show that

a) ijk kji 6ε ε = −

b) kki 0ε =

c) ( )ijk j k ia a 0ε =

51. a) Show that the scalar product ⋅a b of two vectors is invariant under the transformation (rotation) of coordinates. That means that scalar product is a 0th order tensor.

b) Show that Kronecker delta ijδ under the transformation (rotation) of coordinates obeys the tensor rule. That means that ijδ is a 2nd order tensor.

c) Derive the contruction formula (37b).

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 321 52. [RF] Use tensor notation to rewrite:

a) The trace of the matrix A :

( ) 11 22 33tr a a a = + + =A

b) The determinant of the matrix A :

11 22 33 21 32 13 a a a a a a ... = + + =A

53. [RF] Prove the identities:

a) ( ) ( )div grad grad 0ϕ ψ× =

b) ( )curl gradϕ = 0 c) ( )div curl 0=a

54. [RF] Show that any vector field of the form ( )( )( )( )

1 1

1 2 3 2 2

3 3

f xx ,x ,x f x

f x

=

F is irrotational.

55. a) Prove the triple vector product identity ( ) ( ) ( )× × = ⋅ − ⋅a b c a c b a b c

b) Derive the identity for ( ) ( )× × ×a b c d (p.247).

c) Using tensors, prove the identity #11, p.259:

d) Using tensors, prove the identity #15, p.259:

56. [BT] A scalar function ( )ϕ a of a vector argument 1

2 3

3

aaa

= ∈

a is said to be linear if

( ) ( )c c , cϕ ϕ= ∈a a ( ) ( ) ( ) 3, ,ϕ ϕ ϕ+ = + ∈a b a b a b Prove that the most general function of this kind is of the form

( ) 1 2 3a a aϕ α β γ= + +a , where , ,α β γ ∈ .

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 322 57. [D. Maines, Notes], [Bird, Stewart, and Lightfoot “Transport Phenomena”, 1960, SL,p.80,p.119], no solution is given in these references – only formulation of the problem.

The Navier-Stokes equation (governing differential equation for fluid motion with constant density ρ and viscosity μ):

( ) 2P t

ρ µ ρ∂ + ⋅∇ = −∇ + ∇ + ∂ V V V V g

Using tensor notations show that for incompressible, irrotational, steady flow this simplifies to Bernoulli’s equation:

2

P2

ρ ρ

∇ = −∇ +

V g

Given: incompressible ⇒ 0∇⋅ =V (i) irrotational ⇒ 0∇× =V (ii)

steady flow ⇒ 0t

∂=

∂V

(iii)

squared norm 22 2V⋅ = = =V V V V

Hints: 1. Consider identity (14, p.259) with = =a b V and (i). Express ( )⋅∇V V . 2. Consider identity (13, p.259) with =a V and (i),(ii). Express 2 ∇ V .

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 323

Operator nabla and related differential operator ⋅∇v which appears in Nevier-Stokes equations

( ) ( )1 2 3x ,x ,xϕ ϕ=r , ( )( )( )( )

1 1 2 3

2 1 2 3

3 1 2 3

a x ,x ,xa x ,x ,xa x ,x ,x

=

a r , ( )( )( )( )

1 1 2 3

2 1 2 3

3 1 2 3

v x ,x ,xv x ,x ,xv x ,x ,x

=

v r

Operator ∇

1

2

3

x

x

x

∂ ∂ ∂

= ∂ ∂ ∂

ix

∂=∂

Operator ∆ ∇⋅∇ 2 2 2

2 2 21 2 3x x x

∂ ∂ ∂= + +

∂ ∂ ∂

j j

x x∂ ∂

=∂ ∂

( )ϕ∇ ⋅∇ 2 2 2

2 2 21 2 3x x xϕ ϕ ϕ∂ ∂ ∂

= + +∂ ∂ ∂

j j

x x

ϕ∂ ∂=∂ ∂

( )∇ ⋅∇ a

2 2 21 1 1

2 2 21 2 3

2 2 22 2 2

2 2 21 2 3

2 2 23 3 3

2 2 21 2 3

a a ax x x

a a ax x x

a a ax x x

∂ ∂ ∂+ + ∂ ∂ ∂

∂ ∂ ∂ = + +∂ ∂ ∂

∂ ∂ ∂ + + ∂ ∂ ∂

i

j j

ax x∂ ∂

=∂ ∂

Operator ⋅∇v 1 21 2 3

v v vx x x∂ ∂ ∂

= + +∂ ∂ ∂

j

j

vx∂

=∂

( )⋅∇v a

1 1 11 2 3

1 2 3

2 2 21 2 3

1 2 3

3 3 31 2 3

1 2 3

a a av v vx x xa a av v vx x xa a a

v v vx x x

∂ ∂ ∂+ + ∂ ∂ ∂

∂ ∂ ∂= + +

∂ ∂ ∂ ∂ ∂ ∂ + +

∂ ∂ ∂

j i

j

v ax∂

=∂

( )⋅∇v v

1 1 11 2 3

1 2 3

2 2 21 2 3

1 2 3

3 3 31 2 3

1 2 3

v v vv v vx x xv v vv v vx x xv v v

v v vx x x

∂ ∂ ∂+ + ∂ ∂ ∂

∂ ∂ ∂= + +

∂ ∂ ∂ ∂ ∂ ∂ + +

∂ ∂ ∂

j i

j

v vx∂

=∂

( )⋅∇v v = ⋅∇v v ( )= ⋅ ∇v v different notations for ( )⋅∇v v

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 324

( )⋅ ∇v v where ∇v

1 2 31 1 1

1 2 32 2 2

1 2 33 3 3

v v vx x x

v v vx x x

v v vx x x

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

is a dyadic j

i

vx∂

=∂

and operation ( )⋅ ∇v v of vector with a dyadic results in a vector j ij

v vx∂

=∂

( )⋅∇v v ( ) ( )12

= ∇ ⋅ − × ∇×v v v v ( ) ( )21 v2= ∇ − × ∇×v v is a definition of ( )⋅∇v v in BSL (p.726), which is consistent with identity #14 ( )∇ ⋅a b ( ) ( ) ( ) ( )= × ∇× + × ∇× + ⋅∇ + ⋅∇a b b a a b b a for = =a b v ( )∇ ⋅v v ( ) ( )2 2= × ∇× + ⋅∇v v v v

Operator ⋅∇a appears in identity #14

( ) ( ) ( ) ( ) ( )∇ ⋅ = × ∇× + × ∇× + ⋅∇ + ⋅∇a b a b b a a b b a

diadic diadic

= ∇ + ∇a b b a

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 325

https://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=2ahUKEwiAkYH3puHdAhUS2VMKHTxPCn0QjRx6BAgBEAU&url=http://www.wikiwand.com/en/Hydrodynamica&psig=AOvVaw0b4lRSEaFO5te6lh-VVqPB&ust=1538347805822148

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 326 58. [B&T, 246] Electromagnetic field in a medium of dialectric constant ε , magnetic permeability µ , conductivity σ , no free chardge in conducting

medium 0ρ = , and current σ=j E (Ohm’s law) is described by a system of Maxwelll’s equations:

div 0=E Gauss’s law

div 0=H Gauss’s law for magnetism

curl c tµ ∂

= −∂HE Faraday’s law

4curl c t cε πσ∂

= +∂EH E Maxwell-Ampere law

where ( ),t=E E r is the electric field and ( ),t=H H r is the magnetic field. Show that the following wave equation can be derived from Maxwell’s equations:

2

2 2 2

4 tc t c

µε πµσ∆ ∂ ∂= +∂∂

E EE

Hint: apply operator curl to Faraday’s law.

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 327

The monument to Maxwell in Edinburgh near the place ogf his birth.

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 328 Examples of application of Maxwell’s equations:

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 329

From the notes on “Scattering of the laser light by density fluctuations in turbulent flow”

Derivation of the identity #13, p.253 (September 20, 2017)

Chapter IV Vector and Tensor Analysis IV.4 Exercises and Examples August 18, 2020 330