Page 5 - Intro to Tensor Calculus

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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 become 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.

§1.1 INDEX NOTATION
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Two vectors A and B can be expressed in the component form
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A = A 1 b e 1 + A 2 b e 2 + A 3 b e 3 and B = B 1 b e 1 + B 2 b e 2 + B 3 b e 3 ,
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where b e 1 , b e 2 and b e 3 are orthogonal unit basis vectors. Often when no confusion arises, the vectors A and
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B are expressed for brevity sake as number triples. For example, we can write

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A =(A 1 ,A 2 ,A 3 ) and B =(B 1 ,B 2 ,B 3 )
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where it is understood that only the components of the vectors A and B are given. The unit vectors would
be represented
b e 1 =(1, 0, 0), b e 2 =(0, 1, 0), b e 3 =(0, 0, 1).
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A still shorter notation, depicting the vectors A and B is the index or indicial notation. In the index notation,
the quantities
A i , i =1, 2, 3 and B p , p =1, 2, 3

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represent the components of the vectors A and B. This notation focuses attention only on the components of
the vectors and employs a dummy subscript whose range over the integers is specified. The symbol A i refers
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to all of the components of the vector A simultaneously. The dummy subscript i can have any of the integer
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values 1, 2or3. For i = 1 we focus attention on the A 1 component of the vector A. Setting i =2 focuses
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attention on the second component A 2 of the vector A and similarly when i = 3 we can focus attention on
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the third component of A. The subscript i is a dummy subscript and may be replaced by another letter, say
p, so long as one specifies the integer values that this dummy subscript can have.