Chapter 2
Vector and Tensor Analysis
Studies spanning many centuries have shown that mathematics can be used to
describe and predict the behavior of physical systems, sometimes with remarkable
accuracy. Since the behavior of a system does not depend on the specific coordinate
system used to describe it, we should be able to write the governing equations in a
form that is independent of coordinates. Tensor analysis is built on this idea. 1
Physical quantities can be represented by mathematical objects called tensors.
Zeroth-order tensors (scalars) have only a magnitude (e.g., density, temperature),
first-order tensors (vectors) have a magnitude and direction (e.g., velocity, temperature gradient), second-order tensors (dyadics) have a magnitude and two associated
directions (e.g., stress 2 ), and so on.
For illustration, consider a vector a fixed within a given frame of reference A,
which can be in motion relative to other reference frames. Also fixed in A are
two Cartesian coordinate systems, (x, y, z) and ( ¯
x, ¯
y, ¯
z), which differ by a rigidbody rotation of their axes (Fig. 2.1). The vector a can be pictured as an arrow
of specific length (magnitude) pointed in a particular direction. The components
(a x , a y , a z ) relative to the coordinates (x, y, z) can be determined by projecting a
onto the corresponding coordinate axes. Similarly, an alternative set of components
( ¯
a x , ¯
a y , ¯
a z ) for a can be determined by projection onto the ( ¯
x, ¯
y, ¯
z)-axes. Although
the two sets of components are different, the vector a itself is the same in both
coordinate systems (same magnitude and direction) contained in frame A.
This simple example illustrates a fundamental property of vectors. Although
the components of a vector may change, the vector itself is invariant relative to
1 Other introductions to tensor analysis can be found in the excellent books by Flugge (1972),
Malvern (1969), Simmonds (1994), and Holzapfel (2000).
2 A stress component depends on its direction of action and the orientation of the area on which it
acts.
© Springer Nature Switzerland AG 2020
L. A. Taber, Continuum Modeling in Mechanobiology,
https://doi.org/10.1007/978-3-030-43209-6_2
19
Vector and Tensor Analysis
Studies spanning many centuries have shown that mathematics can be used to
describe and predict the behavior of physical systems, sometimes with remarkable
accuracy. Since the behavior of a system does not depend on the specific coordinate
system used to describe it, we should be able to write the governing equations in a
form that is independent of coordinates. Tensor analysis is built on this idea. 1
Physical quantities can be represented by mathematical objects called tensors.
Zeroth-order tensors (scalars) have only a magnitude (e.g., density, temperature),
first-order tensors (vectors) have a magnitude and direction (e.g., velocity, temperature gradient), second-order tensors (dyadics) have a magnitude and two associated
directions (e.g., stress 2 ), and so on.
For illustration, consider a vector a fixed within a given frame of reference A,
which can be in motion relative to other reference frames. Also fixed in A are
two Cartesian coordinate systems, (x, y, z) and ( ¯
x, ¯
y, ¯
z), which differ by a rigidbody rotation of their axes (Fig. 2.1). The vector a can be pictured as an arrow
of specific length (magnitude) pointed in a particular direction. The components
(a x , a y , a z ) relative to the coordinates (x, y, z) can be determined by projecting a
onto the corresponding coordinate axes. Similarly, an alternative set of components
( ¯
a x , ¯
a y , ¯
a z ) for a can be determined by projection onto the ( ¯
x, ¯
y, ¯
z)-axes. Although
the two sets of components are different, the vector a itself is the same in both
coordinate systems (same magnitude and direction) contained in frame A.
This simple example illustrates a fundamental property of vectors. Although
the components of a vector may change, the vector itself is invariant relative to
1 Other introductions to tensor analysis can be found in the excellent books by Flugge (1972),
Malvern (1969), Simmonds (1994), and Holzapfel (2000).
2 A stress component depends on its direction of action and the orientation of the area on which it
acts.
© Springer Nature Switzerland AG 2020
L. A. Taber, Continuum Modeling in Mechanobiology,
https://doi.org/10.1007/978-3-030-43209-6_2
19