Dr. Ahmed G. Abo-Khalil

Electrical Engineering Department

Notation in vector

Vector calculus concerns differentiation and integration of vector or scalar fields particularly in a three-dimensionalEuclidean space, and uses specific notations of differentiation. In a Cartesian coordinate o-xyz, assuming a vector field A is mathbf{A} = (mathbf{A}_x, mathbf{A}_y, mathbf{A}_z), and a scalar field varphi is varphi = f(x,y,z),.

First, a differential operator, or a Hamilton operator  which is called del is symbolically defined in the form of a vector,


abla = left( frac{partial}{partial x}, frac{partial}{partial y}, frac{partial}{partial z} 
ight),

where the terminology symbolically reflects that the operator ∇ will also be treated as an ordinary vector.

φ
  • Gradient: The gradient mathrm{grad,} varphi, of the scalar field varphi is a vector, which is symbolically expressed by the multiplication of ∇ and scalar field varphi,

egin{align}
operatorname{grad} varphi
&= left( frac{partial varphi}{partial x}, frac{partial varphi}{partial y}, frac{partial varphi}{partial z} 
ight) \
&= left( frac{partial}{partial x}, frac{partial}{partial y}, frac{partial}{partial z} 
ight) varphi \
&= 
abla varphi
end{align}

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