Vectors and Tensors R. Shankar Subramanian - Noppa

∂ ∂
∇≡ e + e + e
∂ ∂
() 1 ( 2) ( 3)
∂
∂
x1 x2
x 3
in a rectangular Cartesian coordinate system ( , , )
x x x .
1 2 3
Note that ∇ is an operator and not a vector. So, you should exercise care in manipulating it.
The ∇ operator is the generalization of a derivative. We can differentiate vector fields in more than
one way.
Divergence
∇⋅ v or div v is called the divergence of the vector field v . If the rectangular Cartesian
components of v are vv, v,
then
1, 2 3
∂v
∂v
∂v
1 2 3
∇⋅ v = + +
∂x1 ∂x2
∂x 3
As you can see, the result is a scalar field.
Curl
∇× v or curl v is a vector field. As the name implies, it measures the “rotation” of the vector
v .
Again, in ( x 1,
x 2
, x 3 ) coordinates,
e e e
() 1 ( 2) ( 3)
∂ ∂ ∂
∇× v = ∂ x ∂ x ∂ x
1 2
v v v
1 2
3
3
= ε
ijk
∂v
j
∂x
i
There are two important theorems you should know. They are simply stated here without proof.
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