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[parent] nabla acting on products (Topic)

Let $ f$, $ g$ be differentiable scalar fields and $ \vec{u}$, $ \vec{v}$ differentiable vector fields in some domain of $ \mathbb{R}^3$. There are following formulae:

Explanations

  1. $ \vec{v}\cdot\nabla$ means the operator $ v_x\frac{\partial}{\partial x}+v_y\frac{\partial}{\partial y} +v_z\frac{\partial}{\partial z}$.
  2. The gradient of a vector $ \vec{w}$ is defined as the dyad $ \nabla\vec{w} := \vec{i}\,\frac{\partial\vec{w}}{\partial x} +\vec{j}\,\frac{\partial\vec{w}}{\partial y} +\vec{k}\,\frac{\partial\vec{w}}{\partial z}$.
  3. The divergence and the curl of a dyad product are defined by the equation
    $ \nabla\!*\!(\vec{u}\vec{v}) := \vec{i}\!*\!\frac{\partial(\vec{u}\vec{v})}{\pa... ...vec{v})}{\partial y} +\!\vec{k}\!*\!\frac{\partial(\vec{u}\vec{v})}{\partial z}$, where the asterisks are dots or crosses and the partial derivatives of the dyad product mean the expression $ \frac{\partial(\vec{u}\vec{v})}{\partial x} = \frac{\partial\vec{u}}{\partial x}\vec{v}+ \vec{u}\frac{\partial\vec{v}}{\partial x}$ and so on.



"nabla acting on products" is owned by pahio.
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See Also: $\nabla$, nabla

Also defines:  gradient of vector, divergence of dyad product, curl of dyad product

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Cross-references: expression, partial derivatives, equation, operator, dyad product, dyads, scalar product, vector product, curl, vector, divergence, function, product, gradient, domain, vector fields, fields, scalar, differentiable
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This is version 7 of nabla acting on products, born on 2005-08-06, modified 2005-08-06.
Object id is 7300, canonical name is NablaActingOnProducts.
Accessed 5640 times total.

Classification:
AMS MSC26B10 (Real functions :: Functions of several variables :: Implicit function theorems, Jacobians, transformations with several variables)
 26B12 (Real functions :: Functions of several variables :: Calculus of vector functions)

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