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PART 1: INTRODUCTION TO TENSOR CALCULUS
A scalar field describes a one-to-one correspondence between a single scalar number and a point. An n-
dimensional vector field is described by a one-to-one correspondence between n-numbers and a point. Let us
generalize these concepts by assigning n-squared numbers to a single point or n-cubed numbers to a single
point. When these numbers obey certain transformation laws they e examples of tensor fields. In
general, scalar fields are referred to as tensor fields of rank or order zero whereas vector fields are called
tensor fields of rank or order one.
Closely associated with tensor calculus is the indicial or index notation. In section 1 the indicial
notation is defined and illustrated. We also define and investigate scalar, vector and tensor fields when they
are subjected to various coordinate transformations. It turns out that tensors have certain properties which
are independent of the coordinate system used to describe the tensor. Because of these useful properties,
we can use tensors to represent various fundamental laws occurring in physics, engineering, science and
mathematics. These representations are extremely useful as they are independent of the coordinate systems
considered.
§ INDEX NOTATION
Two vectors A~ and B~ can be expressed in ponent form
A~ = A1 be1 + A2 be2 + A3 be3 and B~ = B1 be1 + B2 be2 + B3 be3,
where be1, be2 and be3 are orthogonal unit basis vectors. Often when no confusion arises, the vectors A~ and
B~ are expressed for brevity sake as number triples. For example, we can write
A~ =(A1,A2,A3)andB~ =(B1,B2,B3)
where it is understood that only ponents of the vectors A~ and B~ are given. The unit vectors would
be represented
be1 =(1, 0, 0), be2 =(0, 1, 0), be3 =(0, 0, 1).
A still shorter notation, depicting the vectors A~ and B~ is the index or indicial notation. In the index notation,
the quantities
Ai,i=1, 2, 3andBp,p=1, 2, 3
represent ponents of the vectors A~ and B.~