Part C Tensor Calculus

Let T=T(t) be a tensor-value function of a scalar t.

0

limt

t t tddt t

T TT

2C1 Tensor-valued functions of a tensor

d d ddt dt dt

T ST S

d d dtdt dt dt

TT T

d d ddt dt dt

T STS S T

d d ddt dt dt

T aTa a T

T

Td ddt dt

TT

Tensors

0

limt

t t t t t tddt t

T a T a

Ta

d d ddt dt dt

T aTa a TShow that:

0

limt

t t t t t t t t t t t tt

T a T a T a T a

0 0

lim limt t

t t t t t tt t t

t t

T T a a

a T

d ddt dt

T aa T

Tensors

Example 2C1.1:Show that in Cartesian coordinates the components of dT/dt, i.e., (dT/dt)ij are given by the derivatives of the components, dTij/dt.

Solution:

ij i jT e Te

31 2 0dd d

dt dt dt

ee e

iji j i j

ij

dT d d ddt dt dt dt

T Te Te e e

Tensors

Example 2C1.2:Show that for an orthogonal tensor Q(t), (dQ/dt)QT is antisymmetric tensor.

solution: T QQ I

0T

Td ddt dt

Q QQ Q

TTd d

dt dt

Q QQ Q

T TT T Td d d d

dt dt dt dt

Q Q Q QQ Q Q Q

T TTd d

dt dt

Q QQ Q

Tensors

Example 2C1.3:A time-dependent rigid body rotation about a fixed point can be represented by a rotation tensor R(t), so that a position vector r0 is transformed through rotation into r(t)=R(t)r0. Derive the equation

Where is the dual vector of the asymmetric tensor .

ddt

r ω r

ω TddtR R

Solution:

0( ) ( )t tr R r

0Td d d

dt dt dt

r R Rr R r ω r

TddtR R is an antisymmetric tensor

is the dual vector of ω

Tensors

TddtR R

2C2 Scalar Field, Gradient of a Scalar FunctionLet be a scalar-valued function of the position vector r .

r

The gradient of at a point r is defined to be vector, denoted by or grad

For the gradient of a scalar, its magnitude is the maximum rate of change at the point; its direction is along the direction where maximum rate of change ocuurs.

Temperature fieldP

Radius △ l

T(P) T(P1) T(P2) T(P3) T(P4) T(P5)10 9.98 9.96 9.94 10.05 10.07

One can obtain the following temperatures after the measurement

lPTPTT i

P

)()(max|

55

( ) ( )|PT P T PT PP

l

Tensors

7

Mathematical definition of gradient

The gradient of scalar field V(x,y,z) at point P0(x0,y0,z0) can be represented as

)(),,()(),,(

)(),,(),,(),,(

00

0000

00

0

zzz

zyxVyyy

zyxV

xxx

zyxVzyxVzyxVV

PP

P

( , , ) ( , , ) ( , , ) ( )V x y z V x y z V x y zV i j k xi yj zkx y z

0 0( , , ) ( , , ) ( , , )V V V x y z V x y z V x y zi j k l V l

l l x y z

Tensors

d d d r r r r

ddr e

dr denotes the magnitude of dre is the unit vector in the direction of dr (note: e=dr/dr)

In the Cartesian coordinate system (e1, e2, e3)

1

1 11

ddr x e

e

2

2 22

ddr x e

e

3

3 33

ddr x e

e

1 2 31 2 3x x x

e e e

Surface of constant

Tensors

Example 2C2.1:If , find a unit vector n normal to the surface of a constant passing through (2,1,0).

1 2 3x x x

Solution:

1 2 3 2 1 1 2 31 2 3

x xx x x

e e e e e e

At the point (2,1,0),1 2 32 e e e

1 2 31 26

n e e e

Tensors

Example 2C2.2:If q denotes the heat flux vector (rate of heat flow/area), the Fourier heat conduction law states that:

where T is the temperature field and k is the thermal conductivity. If find T at A(1,0) and B( , ). Sketch curves of constant T and indicate the vectors q at the two points.

k T q 2 2

1 22T x x 1/ 2

Solutions:

1 2 3 1 1 2 21 2 3

4 4T T TT x xx x x

e e e e e

1 1 2 24k x x q e e

At point A, 14A kq eAt point B, 1 22 2B k q e e 1T

2T 1x

2x

AAq

BqB

Tensors

1/ 2

Example 2C2.3:A more general heat conduction law can be given in the following form:

Where K is a tensor known as thermal conductivity tensor.a). If it is known that K is symmetric, show that there are at least three directions in which heat flow is normal to the surface of constant temperature.b). If T=2x1+3x2 and

find q.

T q K

2 1 01 2 0

0 0 3

K

Solution: （ 1 ） i i ikKn n

1 1 1 1 1T T T k T q K K n Kn n

Note that 1T T n

Tensors

（ 2 ） T=2x1+3x2

2 1 0 2 11 2 0 3 4

0 0 3 0 0

q

1 22 3T e e

1 21 4 q e e

1x

2x q

2T 4T

Tensors

2C3 Vector Field, Gradient of a Vector FieldLet v(r) be a vector-valued function of position r. The gradient of v ( denoted by or grad v) is defined to be the second-order tensor which, when operating on dr gives the difference of v at r+dr and r.

d d d v v r r v r v r

ddr e

v v e

Transforms the unit vector e into the vector describing the rate of change v in that direction.

v

Let dr denotes |dr| and e denote dr/dr

Tensors

v

1

11

ddr x

e

v v v e

In the Cartesian coordinate system (e1, e2, e3),

11 1 1 111

1 1 1

vx x x

vv e v e e e v

j

jj

ddr x

e

v v v e

ii j i iij

j j j

vx x x

vv e v e e e v

Tensors

2C4 Divergence of a Vector Field

31 2

1 2 3

div m

m

v vv vx x x x

v

Let v(r) be a vector field. The divergence of v(r) is defined to be a scalar field given by the trace of the gradient of v.

div tr v v

Tensors

In the Cartesian coordinate system (e1, e2, e3),

n

V

S Definition of Flux

S

SdzyxV

),,(

Closed surface

S

( , , )SV x y z dS

Vd

Gauss Law

Tensors

2C4 Divergence of a Tensor Field

div div -trT T T a T a T a

Let T(r) be a second-order tensor. The divergence of T is defined to be a vector field (div T), such that for any vector a

div -tr div 0T T imi i i i im m

m

Tb Tx

b e T e T e e

div imi

m

Tx

T e

Tensors

In the Cartesian coordinate system (e1, e2, e3), let b=div T,

Example 2C4.1:

If and Show that:

r a a r div div a a a

Solution:Let b a i ib a

div divi ii

i i i

b aa

x x x

b a a

Tensors

Then

and

Example 2C4.2:

Given ， Show that:

r T r div div T T T

Solution:

div ijij i ij i i

j j j

TT T

x x x

T e e e

div imi

m

Tx

T e

div T T

Tensors

2C5 Curl of a Vector Let v(r) be a vector field. The curl of v is defined to be the vector field given by twice the dual vector of the antisymmetric part of , that is v curl 2 Av t

lF

Definition of circulation

( , , )lF x y z dl

F ( , , )

lF x y z dl

S l

F dS F dl

Curl Stokes’ theorem

Tensors

31 2 1

2 1 3 1

31 2 2

2 1 3 2

3 31 2

3 1 3 2

1 102 2

1 102 2

1 1 02 2

A

vv v vx x x x

vv v vx x x x

v vv vx x x x

v

3 32 1 2 11 2 3

2 3 3 1 1 2

curl 2 A v vv v v vx x x x x x

v t e e e

32 1 13 2 21 3A T T T t e e e

TensorsIn the Cartesian coordinate system (e1, e2, e3),

2C6 Laplacian of a Scalar Field Let f(r) be a scalar-valued function of the position vector r. The definition of the Laplacian of a scalar field is given by

2 div trf f f

In rectangular coordinates the Laplacian becomes

2 2 2 2

22 2 21 2 3

tri i

f f f ff fx x x x x

Tensors

2C7 Laplacian of a Vector Field Let v(r) be a vector field. The Laplacian of v is defined by the following:

2 div - v v v

In rectangular coordinates div ki

i k

vx x

v e j

jkk

vx

v e

ji jk i

k

vx x

v e

i jk i jk ij k ik j

jij k ik j i

k

vx x

v e

ii

i

vvx x x x

e

Tensors

2 k ii i

i k i

vv vx x x x x x

v e e

2ii i i

vv

x x

e e

2 2 22 1 1 1

12 2 21 2 3

v v vx x x

v e

2 2 22 2 2

22 2 21 2 3

v v vx x x

e

2 2 23 3 3

32 2 21 2 3

v v vx x x

e

Tensors

Problem1:Consider the scalar field defined by （ a ） Find the unit vectors normal to the surface of constantφat the origin (0,0,0) and point (1,0,1).（ b ） What are the maximum values of the directional derivatives of φ at the origin and point (1,0,1).（ c ） Evaluate at the origin if dr=ds(e1+e2)

21 1 2 33 2x x x x

/d dr

Problem2:Consider the ellipsoid defined by the equation Find the unit normal vector at a given position （ x,y,z ）

2 2 2 2 2 2/ / / 1x a y b z c

Consider the vector field . For the point （ 1,1,0 ）(a)Find the matrix of(b)Find the vector(c)Find(d)If , find the differential dv

2 2 21 1 3 2 2 3x x x v e e e

v v v

div v 1 2 3 / 3d ds r e e e

Tensors

Problem3: