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nabla acting on products
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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:
- Gradient of a product function
$\nabla(fg) = (\nabla f)g+(\nabla g)f$
- Divergence of a scalar-multiplied vector
$\nabla\cdot(f\vec{u}) = (\nabla f)\cdot\vec{u}+(\nabla\cdot\vec{u})f$
- Curl of a scalar-multiplied vector
$\nabla\!\times\!(f\vec{u}) = (\nabla f)\times\vec{u}+(\nabla\!\times\!\vec{u})f$
- Divergence of a vector product
$\nabla\cdot(\vec{u}\!\times\!\vec{v}) = (\nabla\!\times\!\vec{u})\cdot\vec{v}-(\nabla\!\times\!\vec{v})\cdot\vec{u}$
- Curl of a vector product
$\nabla\!\times\!(\vec{u}\!\times\!\vec{v}) = (\vec{v}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{v} -(\nabla\cdot\vec{u})\vec{v}+(\nabla\cdot\vec{v})\vec{u}$
- Gradient of a scalar product
$\nabla(\vec{u}\cdot\vec{v})\, =\, (\vec{v}\cdot\nabla)\vec{u}+(\vec{u}\cdot\nabla)\vec{v} +\vec{v}\!\times\!(\nabla\!\times\!\vec{u}) +\vec{u}\!\times\!(\nabla\!\times\!\vec{v})$ or, using dyads,
$\nabla(\vec{u}\cdot\vec{v}) = (\nabla\vec{u})\cdot\vec{v}+(\nabla\vec{v})\cdot\vec{u}$
- Gradient of a vector product
$\nabla(\vec{u}\!\times\!\vec{v}) = (\nabla\vec{u})\!\times\!\vec{v}-(\nabla\vec{v})\!\times\!\vec{u}$
- Divergence of a dyad product
$\nabla\cdot(\vec{u}\,\vec{v}) = (\nabla\!\cdot\!\vec{u})\,\vec{v}+\vec{u}\cdot\nabla\vec{v}$
- Curl of a dyad product
$\nabla\!\times\!(\vec{u}\,\vec{v}) = (\nabla\!\times\!\vec{u})\,\vec{v}-\vec{u}\times\!\nabla\vec{v}$
Explanations
- $\vec{v}\cdot\nabla$ means the operator $v_x\frac{\partial}{\partial x}+v_y\frac{\partial}{\partial y} +v_z\frac{\partial}{\partial z}$
- The gradient of a vector $\vec{w}$ is defined as the dyad $\nabla\vec{w} := \vec{i}\,\frac{\partial\vec{w}}{\partial x} +\vec{j}\,\frac{\partial\vec{w}}{\partial y} +\vec{k}\,\frac{\partial\vec{w}}{\partial z}$
- The divergence and the curl of a dyad product are defined by the equation
$\nabla\!*\!(\vec{u}\vec{v}) := \vec{i}\!*\!\frac{\partial(\vec{u}\vec{v})}{\partial x} \!+\!\vec{j}\!*\!\frac{\partial(\vec{u}\vec{v})}{\partial y} \!+\!\vec{k}\!*\!\frac{\partial(\vec{u}\vec{v})}{\partial z}$ 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.
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"nabla acting on products" is owned by pahio.
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See Also: , nabla
| Also defines: |
gradient of vector, divergence of dyad product, curl of dyad product |
This object's parent.
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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 8 of nabla acting on products, born on 2005-08-06, modified 2008-10-08.
Object id is 7300, canonical name is NablaActingOnProducts.
Accessed 9152 times total.
Classification:
| AMS MSC: | 26B10 (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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Pending Errata and Addenda
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