Introduction  (Page 2/5)

Scalar, vector and tensor fields

Scalars, vectors, and matrices are concepts that may have been introduced to the student in a course in linear algebra. Here, scalar, vector, and tensor fields are entities that are defined over some region of 3-D space and time. It is implicit that they are a function of the spatial coordinates and time, i.e., $\phi =\phi (x,y,z,t)=\phi (x,t)$ . The spatial coordinates are expressed as Cartesian coordinates in this class. However, vectors and tensors are physical entities that are independent of the choice of spatial coordinates even though their components depend on the choice of coordinates.

Scalar fields have a single number, a scalar, at each point in space. An example is the temperature of a body. The temperature field is usually expressed visually by a contour map showing curves of constant temperature or isotherms. An alternative visual display of a scalar field is a color map with the value of the scalar scaled to a gray scale, hue, or saturation. The values of the scalar field are continuous with the exception of definable surfaces of discontinuity. An example is the density of two fluids separated by an interface. Media that are chaotic and discontinuous on a microscopic scale may be described by an average value in a representative elementary volume that is large compared to the microscopic heterogeneity but small compared to macroscopic variations. An example is the porosity of a porous solid.

Vector fields have a magnitude and direction associated with each point in space. An example is the velocity field of a fluid in motion. Vector fields in two dimensions can be visually expressed as field lines that are everywhere tangent to the vector field and whose separation quantifies the magnitude of the field. Streamlines are the field lines of the velocity field. Alternatively, a vector field in two dimensions can be visually expressed by arrows whose directions are parallel to the vector and having a width and/or length that scales to the magnitude of the vector. These graphical representations of vector fields are not useful in three dimensions. In general, a vector field in 3-D can be expressed in terms of its components projected on to the axis of a coordinate system. Thus, a vector field may have different components when projected on to different coordinate systems. Since a vector is a physical entity, the components in different frames of reference transform by prescribed rules. The position of a point in space relative to an origin is a vector defined by the distance and direction. Special vectors having a magnitude of unity are called unit vectors and are used to define a direction such as coordinate directions or the normal direction to a surface. We will denote vectors with bold face letters, e.g., v, x, or n .

Tensors are physical entities associated with two directions. For example, the stress tensor represents the force per unit area, each of which are directional quantities. The velocity gradient is a tensor. Transport coefficients, such as the thermal conductivity, are tensors, which transform a potential gradient to a flux, each of which are vectors. The components of a tensor in a particular coordinate system are represented by a 3×3 matrix. Since the tensor is a physical entity that is independent of the coordinate system, the components must satisfy certain transformation rules between coordinate systems. In particular, a set of three directions called the principal directions can be found to transform the components of the tensor to a diagonal matrix. This corresponds to finding the eigenvectors of a matrix and the components correspond to the eigenvalues. Bold face letters will also denote tensors. The stress tensor will be denoted by T or τ , depending on whether discussing Aris or BSL, respectively.