nabla acting on products
Let f, g be differentiable scalar fields and →u, →v differentiable vector fields in some domain of ℝ3. There are following formulae:
-
•
Gradient
of a product function
∇(fg)=(∇f)g+(∇g)f -
•
Divergence
of a scalar-multiplied vector
∇⋅(f→u)=(∇f)⋅→u+(∇⋅→u)f -
•
Curl of a scalar-multiplied vector
∇×(f→u)=(∇f)×→u+(∇×→u)f -
•
Divergence of a vector product
∇⋅(→u×→v)=(∇×→u)⋅→v-(∇×→v)⋅→u -
•
Curl of a vector product
∇×(→u×→v)=(→v⋅∇)→u-(→u⋅∇)→v-(∇⋅→u)→v+(∇⋅→v)→u -
•
Gradient of a scalar product
∇(→u⋅→v)=(→v⋅∇)→u+(→u⋅∇)→v+→v×(∇×→u)+→u×(∇×→v)
or, using dyads,
∇(→u⋅→v)=(∇→u)⋅→v+(∇→v)⋅→u -
•
Gradient of a vector product
∇(→u×→v)=(∇→u)×→v-(∇→v)×→u -
•
Divergence of a dyad product
∇⋅(→u→v)=(∇⋅→u)→v+→u⋅∇→v -
•
Curl of a dyad product
∇×(→u→v)=(∇×→u)→v-→u×∇→v
Explanations
-
1.
→v⋅∇ means the operator vx∂∂x+vy∂∂y+vz∂∂z.
-
2.
The gradient of a vector →w is defined as the dyad ∇→w:=.
-
3.
The divergence and the curl of a dyad product are defined by the equation
, where the asterisks are dots or crosses and the partial derivativesof the dyad product the expression and so on.
Title | nabla acting on products |
Canonical name | NablaActingOnProducts |
Date of creation | 2013-03-22 15:27:05 |
Last modified on | 2013-03-22 15:27:05 |
Owner | pahio (2872) |
Last modified by | pahio (2872) |
Numerical id | 11 |
Author | pahio (2872) |
Entry type | Topic |
Classification | msc 26B12 |
Classification | msc 26B10 |
Related topic | Nabla |
Related topic | NablaNabla |
Defines | gradient of vector |
Defines | divergence of dyad product |
Defines | curl of dyad product |