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2 Vector and Tensor Analysis
Table 2.1 Vector and dyadic
formulas
a · b = b · a
a · (b + c) = a · b + a · c
a(b + c) = ab + ac
a × b = −b × a
a × (b × c) = (a · c)b − (a · b)c = a · (cb − bc)
(a × b) × c = (a · c)b − (b · c)a = c · (ab − ba)
a · (b × c) = b · (c × a) = c · (a × b)
Table 2.2 Tensor formulas
T · (U + V) = T · U + T · V
T · (U · V) = (T · U) · V
T : U = T T : U T = U : T
T : (U + V) = T : U + T : V
T : (U · V) = U : (T · V T ) = V : (U T · T)
Computation with vectors and tensors is sometimes easier using matrix algebra.
For this purpose, we can write T in the matrix form
T = [T ij ] =
⎡
⎣
T 11 T 12 T 13
T 21 T 22 T 23
T 31 T 32 T 33
⎤
⎦ ,
(2.17)
where it is understood that position (ij ) corresponds to the dyad e i e j (i = row, j =
column). Since the matrix components do not explicitly include the base vectors, it
is important to make sure that the bases are consistent. For example, multiplying a
matrix containing Cartesian components with one that contains cylindrical components would produce nonsense. Such is not the case in manipulating dyadics, where
the base vectors are explicitly included. Hence, one way to check for consistency is
to use dyadic notation to carry out a small part of the calculation.
Tables 2.1 and 2.2 contain some useful relations for vectors, dyadics, and tensors.
2.4 Some Properties of Tensors
Several properties of second-order tensors follow from their matrix representations.
2.4.1 Transpose, Trace, and Determinant
The transpose of a matrix is obtained by interchanging rows and columns, which is
the same as switching the order of the indices in its components. Analogously, the
2 Vector and Tensor Analysis
Table 2.1 Vector and dyadic
formulas
a · b = b · a
a · (b + c) = a · b + a · c
a(b + c) = ab + ac
a × b = −b × a
a × (b × c) = (a · c)b − (a · b)c = a · (cb − bc)
(a × b) × c = (a · c)b − (b · c)a = c · (ab − ba)
a · (b × c) = b · (c × a) = c · (a × b)
Table 2.2 Tensor formulas
T · (U + V) = T · U + T · V
T · (U · V) = (T · U) · V
T : U = T T : U T = U : T
T : (U + V) = T : U + T : V
T : (U · V) = U : (T · V T ) = V : (U T · T)
Computation with vectors and tensors is sometimes easier using matrix algebra.
For this purpose, we can write T in the matrix form
T = [T ij ] =
⎡
⎣
T 11 T 12 T 13
T 21 T 22 T 23
T 31 T 32 T 33
⎤
⎦ ,
(2.17)
where it is understood that position (ij ) corresponds to the dyad e i e j (i = row, j =
column). Since the matrix components do not explicitly include the base vectors, it
is important to make sure that the bases are consistent. For example, multiplying a
matrix containing Cartesian components with one that contains cylindrical components would produce nonsense. Such is not the case in manipulating dyadics, where
the base vectors are explicitly included. Hence, one way to check for consistency is
to use dyadic notation to carry out a small part of the calculation.
Tables 2.1 and 2.2 contain some useful relations for vectors, dyadics, and tensors.
2.4 Some Properties of Tensors
Several properties of second-order tensors follow from their matrix representations.
2.4.1 Transpose, Trace, and Determinant
The transpose of a matrix is obtained by interchanging rows and columns, which is
the same as switching the order of the indices in its components. Analogously, the