Mathematical Conventions

• Where possible, vector quantities are displayed with a raised arrow (e.g., ${\vec a}$, ${\vec A}$). Boldfaced characters are reserved for vectors and matrices as they apply to linear algebra (e.g., the identity matrix, ${\bf I}$).

• The operator $\nabla$, referred to as grad, nabla, or del, represents the partial derivative of a quantity with respect to all directions in the chosen coordinate system. In Cartesian coordinates, $\nabla$ is defined to be

  
  

  
  
  

  $\nabla$ appears in several ways:

  • The gradient of a scalar quantity is the vector whose components are the partial derivatives; for example,

    
    

    
    
    

  • The gradient of a vector quantity is a second-order tensor; for example, in Cartesian coordinates,

    
    

    
    
    

    This tensor is usually written as

    
    

    
    
    

  • The divergence of a vector quantity, which is the inner product between $\nabla$ and a vector; for example,

    
    

    
    
    

  • The operator $\nabla \cdot \nabla$, which is usually written as $\nabla^2$ and is known as the Laplacian; for example,

    
    

    
    
    

    $\nabla^2 T$ is different from the expression $(\nabla T)^2$, which is defined as

    
    

    
    
    

• An exception to the use of $\nabla$ is found in the discussion of Reynolds stresses in Chapter  12, where convention dictates the use of Cartesian tensor notation. In this chapter, you will also find that some velocity vector components are written as $u$, $v$, and $w$ instead of the conventional $v$ with directional subscripts.