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