Basics of Tensors: An Attempt of Making Sense of Tensors
First post of April 2017. No joke.
Introduction
Trying to
finally make some sense of tensor calculus, here is what I learned so far. I still consider myself a novice when it
comes to tensor calculus theory. In
tensors, both subscripts and superscripts are used as indices. To be honest, that throws me off since I’m
used to seeing superscripts as symbols for power.
When I first
learned about tensors, I was tensors were just matrices. It turns out, matrices are a subset of
tensors and whether we’ve been knowing it or not, we use tensors every day in
mathematics: scalars (numbers), vectors, and numbers. In a way, scalars can be thought of as
0-dimensional tensors (points), vectors are 1-dimensional tensors, and matrices
are 2-dimesnional tensors.
To maintain
clarity, I will use strictly subscripts, followed by underscores ( _ ).
The General Definition of
Tensors
Einstein
Sums: Note that a and x may not
necessarily be scalars, but also functions.
Σ a_i * x_i
from i=1 to n = a_1*x_1 + a_2*x_2 + a_3*x_3 + … + a_n*x_n
In tensor
calculus, the sum is simply expressed as a_i
* x_i.
Double
Sums:
a_i,j
= Σ a_i,j * x_i
* y_j (sum over i)
= Σ a_i,j * x_i
* y_j (sum over j)
On a personal note, I prefer using the Σ
symbol, only for clarity. Because
without context a_n*x_n to me a single term and not representing a sum.
Kronercker
Delta: fundamental function in tensor calculus
δ_i, j = { 1 if
i=j; 0 if i≠j
Tensor: A nth
rank tensor in m-dimensional space
that has n indices m^n components of each tensor. The components can be numbers or functions.
Included in the
family of tensors are:
Scalars: 0 indices
Vectors: 1
index
Matrices: 2 indices
Matrices in Tensor
Notation
The notation [
a_i,j ]_m,n is a matrix with dimensions m
* n with indices i and j for each components.
Multiplication
of Matrices A and B with:
A = [a_i,j]_m,n
B = [b_i,j]_n,k
AB = [ a_i,r *
b_r,j ] _m,k
Identity
Matrix:
I = [ δ_i,j
]_n,n
Transpose
Matrix:
A^T = [ a_j,I
]_n,m
Determinant:
det A = e_i,1 *
e_i,2 * e_i,3 * … * e_i,n * a_1_i,1 * a_2_i,2 * a_3_i,3 * … * a_n_i,n
where e is the
Levi-Civita symbol with represents a sign of permutations of numbers 1, 2, 3,
…, n
In general,
e = {
+1 for even permutations (even number of
two-number swaps),
-1 for odd
permutations (odd number of two-number swaps),
0 otherwise
= Π sign(a_j –
a_i) for 1≤i<j≤n
= sign(a_2 –
a_1) * sign(a_3 – a_1) * … * sign(a_n – a_1) * sign(a_3 – a_2) * sign(a_4 –
a_2) * … * sign(a_n – a_2) * … *
sign(a_n – a_n-1)
For 2
dimensions:
e_i,j = { +1
for i=1, j=2; -1 for i=2, j=1, 0 for i=j
For 3 dimensions:
e_i,j,k = { + 1
for permutations (1,2,3), (2,3,1), (3,1,2),
-1 for
permutations (3,2,1), (1,3,2), (2,1,3), 0 for other permutations
Contravariant and Covariant Transformations
Let T_i and W_i
be vector fields with systems x_i and y_i respectively. For the transformation T to W:
Contravariant
Vector Transformation: W_i = T_r * (δy_i
/ δx_r)
Covariant
Transformation: W_i = T_r * (δx_r /
δy_i)
The difference
between the two transformations is in the partial deritival and what variable
gets differentiated in respect to its corresponding function. Note that W and T are assumed to contain
functions.
Cartesian and Affine
Tensors
Cartesian
Tensors: A tensor in 3-dimensional Euclidian
space ([x, y, z]). With Cartesian
tensors, there is no distinction between covariant and contravariant indices.
Affine
Tensors: Affine tensors are a
specialized version of Cartesian tensors formed by:
T: y_i = a_i,j
* x_j (in the context of sums,
hence: T: y_i = Σ a_i,j * x_j)
Where det a_i,j
≠ 0. In an affine tensor, x and y are considered functions.
In an affine transformation, lines that are parallel are preserved.
The Jacobian of
an affine tensor is:
J = [ [ ∂y_1/∂x_1
… ∂y_1/∂x_n ] … [ ∂y_n/∂x_1 … ∂y_n/∂x_n] ]
And it’s
inverse:
J^-1 = [ [ ∂x_1/∂y_1
… ∂x_1/∂y_n ] … [ ∂x_n/∂y_1 … ∂x_n/∂y_n] ]
Metric Tensors
Metric
Tensor: type of function which takes a
pair of tangent vector V and W with scalars g which familiarizes dot products.
In the following,
x_i is assumed to be a function of an
independent variable, like t.
Arc
Length:
ds^2 = δ_i,j *
dx_i * dx_j = Σ (dx_i/dt)^2
Norm:
||V|| = √(V^2)
= √(Σ v_i * v_i)
Angle Between
Vectors:
cos θ = (U * V)
/ (||U|| * ||V||)
Area:
∫ ∫ | R_u x
R_v| du dv
That is some of
the basics of tensor calculus.
Source: Kay, David C. Ph. D. Schaum’s Outlines: Tensor Calculus McGraw Hill.
New York, 2011
Eddie
This blog is
property of Edward Shore, 2017