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Let $f:\mathbb{R}^n\to \mathbb{R}$ be a $C^1(\mathbb{R}^n)$ function, that is, a partially differentiable function in all its coordinates. The symbol $\nabla$ , named nabla, represents the gradient operator, whose action on $f(x_1,x_2,\ldots,x_n)$ is given by \begin{eqnarray*} \nabla f&=&\left(f_{x_1},f_{x_2},\ldots,f_{x_n}\right)\\ &=&\left( \frac{\partial f}{\partial x_1},\frac{\partial f}{\partial x_2},\ldots,\frac{\partial f}{\partial x_n} \right) \end{eqnarray*}
- If $f,g$ are functions, then$$ \nabla(fg) = (\nabla f) g + f \nabla g.$$
- For any scalars $\alpha$ and $\beta$ and functions $f$ and $g$ ,$$ \nabla(\alpha f + \beta g) = \alpha \nabla f + \beta \nabla g.$$
Using the $\nabla$ formalism, the divergence operator can be expressed as $\nabla\cdot$ , the curl operator as $\nabla\times$ , and the Laplacian operator as $\nabla^2$ . To wit, for a given vector field$$ \vA = A_x\, \vi + A_y\, \vj + A_z\, \vk,$$ and a given function $f$ we have
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"nabla" is owned by stevecheng. [ full author list (4) | owner history (4) ]
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Cross-references: vector field, Laplacian, curl, divergence, Formalism, scalars, action, operator, gradient, represents, coordinates, differentiable function, function
There are 3 references to this entry.
This is version 4 of nabla, born on 2003-10-15, modified 2005-10-09.
Object id is 4847, canonical name is NablaNabla.
Accessed 7019 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) |
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Pending Errata and Addenda
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