Math Review We are already familiar with the concept of a scalar and vector. Example: Position (x, y) in two dimensions 1 2 2 2 s ( x y ) where s is the length of x ( xy, ) xi yj And i, j ii 1 j j 1 i j 0 are unit vectors in the x and y directions respectively with We also write x j, j 1, 2 or x j = 1,2,3 Matrices are two component ojects with the same numer of rows and columns
a a a a a a. a A.... a a. a 11 12 13 1n 21 22 2n n1 n2 nn A ij ( row, column) Concept of vector Magnitude V is a Vector V is the magnitude of a three dimensional vector V ( V V V ) 1 2 2 2 2 x y z Note that sometimes the suscripts 1,2,3 are used for x,y,z. Matrix Determinant a a a A det a a a a a a 11 12 13 21 22 23 31 32 33 A a ( a a a ) 23 11 22 33 23a
a 12 ( a21a33 a23a31) 31( a12a23 a22a13 a ) Dot product of two vectors (n=3) i i i i i1 n A A A A A Acos( ) 1 1 2 2 3 3 Repeated indices indicate summation Matrix Multiplication Cij A Aik k j 2D Example a a 11 21 a a 12 22 11 21 12 22 ( a ( a 11 21 11 11 a a 12 22 21 21 ) ) ( a ( a 11 21 12 12 a a 12 22 22 22 ) ) Properties of the determinant of a Matrix A det(a) A A Trace Tr( A) n A ii i1 No similar rule for trace Matrix Inverse 1 ( ) ij jk jk A A
Where jk is the unity ( sometimes called Kronecker delta) matrix with ij = 1 for j = k = 0 for j k 1 0 0 1 Another useful matrix operation is the transpose defined y A T ij A ji characterized y the rows and columns shifted. Concept of Tensors Ojects such as scalars, vectors, matrices and others such as ijk (a third order oject re indices) can undergo transformations (not defined yet) if these ojects remain as the same type we call these tensors. The transformation that we will consider is rotation We see that q ' q We can also look at this prolem a it differently
Exponential Notation exp(i cos( i sin( exp(i 1 (Imaginary Axis) i sin( cos( (Real Axis) Note is in radians. 2 radians = 360 o Oscillatory notation exp(it the angular frequency,t time Derivative properties d(exp(x) dx exp(x) Solution of Equations 2 df d f 2 qf 0 s f 0 2 dx dx f A exp( qx) f A exp( sx) exp(sx)
since 1 T ( R ) ij R jk Rkj RR ij kj ik Note: det ( R ) = 1 follows from the aove equation and det( ) 1 Physical Application of scalars, vectors, 2 nd order tensors: Scalars: Temperature, Pressure, Length, Speed Vectors: Force, Velocity, and Acceleration 2 nd Order Tensors: Stress tensor, Rate of Strain Tensor Concept of Stress Stress is a force per unit area A Given some area, A which is sufficiently small to e considered a plane surface. We define the vector A as the magnitude of the area A directed normal to the surface. A is now a vector! We sometimes write
A An Where A is the magnitude of the area and n is a unit vector in the direction of A Stress is defined as force applied on a particular vector area. We will show that stress is a second order tensor. Shown elow is a figure of Cohen and Kundu. Note the directions of the 1,2,3 directions. We define ij as the force in the on area i in the j direction. See Figure elow. In three dimensions we can write the stress tensor as (in matrix form) 11 12 13 21 22 23 31 32 33 Note that the force on an area A is given y
R 1 1 2 2 1 1 2 2 Which represents a rotation of 45 o! Each column of R represent the two eignefunctions. Math Operation We have already discussed the dot product Xi Yi X Y XYcos X Y 3 i1 X Y i i Note that the dot product of two vectors yields a scalar. The Cross Product of two vectors yields another vector. Ax C X C A sin
C is directed perpendicular to the plane of A, y the right hand screw rule and points out of the paper, which is indicated y the x. Levi-Cevita Symol ijk also called the Alternating Tensor. It is a third order tensor! One of its main usage is in defining the cross product. examples ijk 1 for i, j, k in sequence -1 for i, j, k not in sequence 0 if any two indices are repeated 1 123 312 231 1 321 132 213 An imp[ortant relationship (qwhich is also a homework prolem is ijk klm ie jm im je One can write C Ax C A i ijk j k Vector Calculus We want to generalize some one-dimensional calculus ideas to 2 and 3 dimensions F=F(x)
df F Fx dx dx If F(x) =0, x - F x x df dx dx Sometimes a function has more than one variale F(x, y, z, t) Which we can write as F xt, Consider: F x i Where i 1, 2, 3 x x x 1 2 3 x y z F x 1 F x Implied is y,z, are held constant. F F F( x x, y, z) F( x, y, z) lim x 0 x x x 1 G i F x i F F F G F i j k x x x Curl of a vector
xh xh,take derivation in direction perpendicular to H and then form a vector component perpendicular to oth of them/ C xh H k ijk x j Consider some function x, yz, The equation x, yz, =C C is a constant defines a surface in three dimensions. Solve for say z = F[x,y;C] We can write i j k x x x However if Thus 0 x, yz, were constant then
Concept of Dot product A A cos( ) A A A A i i 1 1 2 2 3 3 2 axis A A 1 1 A 2 1 axis 2 Repeated indices are summed!
Concept of Cross Product 2 axis C=A A sin( ) A 1 123 231 312 1 213 321 132 ijk j k C points out of the oard A A 1 1 C A A 3 1 2 2 1 A 2 1 axis 2
C=A A sin( ) A 1 123 231 312 1 213 321 132 ijk j k 111 121 112 122 131 113 133 0 There are 14 other cominations which are zero. C=A A 1 2 3 3 2 C=A A 2 3 1 1 3 C=A A 3 1 2 2 2
Dirac Delta Function ij 1 0 0 0 1 0 0 0 1 1 0 0a1 a1 a 0 1 0 a a a 0 0 1a a ij j 2 2 i 3 3 a ij j i ij i j a aa a a a 1 1 2 2 3 3 What is? ijk jk?