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Appendix A
VECTORS, TENSORS AND
MATRIX NOTATION
The objective of this section is to review some of the vector operations that you have already covered
in your MATH and ENGR courses. For more details and examples you should refer to your calculus
text under the chapters on vectors and vector calculus.
The following notes will be divided into two parts. The ±rst will review the theory, while the
second part will review how to perform various vector operations in Scienti±c Workplace. These
calculations will be illustrated in Cartesian rectangular and cylindrical coordinates. First you will
learn the basic operations, then you will be shown some examples, and ±nally you will work some
problems.
A.1 Review of Vector and Matrix Operations
In Engineering, we represent physical quantities using three di²erent groups of mathematical ob
jects, i.e., scalars, vectors and tensors. A scalar quantity is represented by a real number with some
appropriate units (mass, temperature, energy, time, etc.). A vector is an object that has a scalar
magnitude and a direction. A vector in threedimensional space can be described as a linear com
bination of three base vectors that have unit length and point in the positive direction of the three
axes; these form the socalled
standard orthonormal basis
. They are denoted
i
,
j
, and
k
and
point parallel to the
x
,
y
, and
z
axes, respectively. Using them, we can write a threespace vector
a
in the following form
a
=
a
1
i
+
a
2
j
+
a
3
k
(A.1)
where
a
1
,
a
2
,
a
3
are the scalar components of the vector
a
with respect to the standard orthonormal
basis.
Note that in printed text, lowercase roman boldface letters are generally used to represent
vectors, and subscripted lowercase italic letters represent their components. In handwriting, the
vector
a
is often written as ¯
a
,
~a
or
a
∼
.
A given vector
a
can be expressed in matrix form as a 3
×
1 column matrix whose entries are the
components of the vector:
[
a
]=
a
1
a
2
a
3
(A.2)
357
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APPENDIX A. VECTORS, TENSORS AND MATRIX NOTATION
We normally think of a vector as a column matrix, but a vector may also be written in matrix
notation as a 1
×
3row matrix:
[
a
]=
£
a
1
a
2
a
3
¤
(A.3)
Addition of vectors is deFned componentwise by
(
a
+
b
)
i
=
a
i
+
b
i
for all
i
.
(A.4)
Multiplication of a vector by a scalar is deFned componentwise by
(
c
a
)
i
=
c
·
a
i
for all
i
.
(A.5)
The di±erence
a
−
b
is simply
a
+(
−
1)
b
. Analogous deFnitions hold for general matrices.
The above deFnitions arise from their geometrical usefulness and from obvious analogy to oper
ations on the real numbers. How to deFne a useful form of multiplication of one vector by another
is not so obvious. We deFne three products of vectors: the
dot product
(or scalar product), the
cross product
(or vector product) and the
dyadic product
(or tensor product). All are products
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 Spring '11
 Wereley

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