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Del Operator: Gradient, Divergence, and Curl, Summaries of Mathematics

Aimst University Mathematics

The del operator is a vector calculus symbol used for compact notation. It is defined as the gradient operator (∇) which is the sum of partial derivatives with respect to x, y, and z axes. The gradient of a function results in a vector, while the divergence of a vector yields a scalar. The curl of a vector is expressed as the cross product of the gradient operator and the vector, resulting in a vector.

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Del Operator

The symbolic operator ”del” is widely used for compact notation. It is a vector defined as

∇=∂

∂x i+∂

∂y j+∂

∂z k= ( ∂

∂x ,∂

∂y ,∂

∂z ) (1)

where i,j,kcorrespond to the unit vector in the x,y,zaxis.

Recall that the product of ∂

∂x and a function f(x, y, z ) is ∂ f

∂x . Then we have the gradient of a

function fexpressed as

grad f=∇f= (∂f

∂x ,∂f

∂y ,∂f

∂z ) (2)

The gradient of a function fis a vector.

Moreover, for a vector F= (u, v, w) where u=u(x, y, z), v=v(x, y , z), w=w(x, y, z) are all

functions of x,y,z, the divergence of this vector Fcan be seen as the dot product of ∇and vector

F, expressed as

div f=∇ · F=∂u

∂x +∂v

∂y +∂w

∂z (3)

The divergence of a vector results in a scalar.

On the other hand, the curl of the vector Fcan be consideres as the cross product of ∇and

vector F, expressed as

curl f=∇ × F=



i j k

∂

∂x

∂

∂y

∂

∂z

u v w



=i(∂w

∂y −∂v

∂z ) + j(∂u

∂z −∂w

∂x ) + k(∂v

∂x −∂u

∂y ) (4)

Note that the curl of a vector results in a vector. Equation (4) can also be expressed as

curl f=∇ × F=



i j k

∂

∂x

∂

∂y

∂

∂z

u v w



= [(∂w

∂y −∂v

∂z ),(∂u

∂z −∂w

∂x ),(∂v

∂x −∂u

∂y )] (5)

1

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Del Operator

The symbolic operator ”del” is widely used for compact notation. It is a vector defined as

∂x

i + ∂ ∂y

j + ∂ ∂z

k = ( ∂ ∂x

∂y

∂z

where i, j, k correspond to the unit vector in the x, y, z axis.

Recall that the product of (^) ∂x∂ and a function f (x, y, z) is ∂f∂x. Then we have the gradient of a

function f expressed as

grad f = ∇f = ( ∂f ∂x

, ∂f ∂y

, ∂f ∂z

The gradient of a function f is a vector.

Moreover, for a vector F = (u, v, w) where u = u(x, y, z), v = v(x, y, z), w = w(x, y, z) are all

functions of x, y, z, the divergence of this vector F can be seen as the dot product of ∇ and vector

F, expressed as

div f = ∇ · F = ∂u∂x + (^) ∂y∂v + ∂w∂z (3)

The divergence of a vector results in a scalar.

On the other hand, the curl of the vector F can be consideres as the cross product of ∇ and

vector F, expressed as

curl f = ∇ × F =

i j k

∂ ∂x

∂ ∂y

∂ ∂z

u v w

= i( ∂w ∂y

− ∂v ∂z

) + j( ∂u ∂z

− ∂w ∂x

) + k( ∂v ∂x

− ∂u ∂y

Note that the curl of a vector results in a vector. Equation (4) can also be expressed as

curl f = ∇ × F =

i j k

∂ ∂x

∂ ∂y

∂ ∂z

u v w

= [( ∂w ∂y

− ∂v ∂z

), ( ∂u ∂z

− ∂w ∂x

), ( ∂v ∂x

− ∂u ∂y

Del Operator: Gradient, Divergence, and Curl, Summaries of Mathematics

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Del Operator

)] (5)