Vector Calculus

Definitions
Wikipedia list of defintions for del.

Identities
Wikipedia list of vector calculus identities.
 * $$\nabla\times(\nabla\times\bold{v}) = \nabla(\nabla\cdot\bold{v}) - \nabla^2\bold{v}$$
 * "curl curl equals grad div take laplacian"

Divergence
The divergence of a vector field can be thought of as the overall flux of the vector field going through an infinitely tiny box at some point. If you imagine some closed 3D surface, say a sphere of some size, with a charge at the centre of it. The field lines coming off of the charge and going through the surface of the sphere is considered it's flux. If you shrink this surface down infinitely, you end up with the flux going out of some point surrounded by the surface. This represents the divergence.

The proof is much more convincing, and if anyone can come up with a better word-based explanation go right ahead. OwenGage 23:20, January 25, 2012 (UTC)

Constant Vector
For a contstant vector,


 * $$\bold{v} = a\hat{\bold{i}}+b\hat{\bold{j}}+c\hat{\bold{k}}$$,

the dot product will be zero as it takes the gradient of each component.


 * $$\nabla\cdot\bold{v} = \frac{\partial}{\partial x}a + \frac{\partial}{\partial y}b + \frac{\partial}{\partial z}c = 0$$

Non-constant Vector
For a vector such as


 * $$\bold{v} = ax\hat{\bold{i}} + by\hat{\bold{j}} + cz\hat{\bold{k}}$$,

the dot product produces scalar value.


 * $$\nabla\cdot\bold{v} = a+b+c$$

General Vector Field
For a vector field


 * $$\bold{v} = \phi_1 \hat{\bold{i}} + \phi_2 \hat{\bold{j}} + \phi_3 \hat{\bold{k}} $$

where


 * $$\phi_n = \phi_n(x,y,z) \,$$,


 * $$ \nabla\cdot\bold{v} = \frac{\partial \phi_1 }{\partial x}\hat{\bold{i}} + \frac{\partial \phi_2 }{\partial y}\hat{\bold{j}} + \frac{\partial \phi_3 }{\partial z}\hat{\bold{k}}$$

It is important to note that the functions $$\phi_1\,$$, $$\phi_2\,$$ and $$\phi_3\,$$, and therefore their derivatives, can all contain x, y and z.