## 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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