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IEVref: | 102-05-19 | ID: | |

Language: | en | Status: Standard | |

Term: | gradient | ||

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Definition: | vector grad f associated at each point of a given space region with a scalar f, having a direction normal to the surface on which the scalar field has a constant value, in the sense of increasing value of f, and a magnitude equal to the absolute value of the derivative of f with respect to distance in this normal directionNote 1 to entry: The gradient expresses the variation of the scalar field quantity from the given point to a point at an infinitesimal distance $\mathrm{d}s$ in the direction of a given unit vector f = grad f ⋅ des. Note 2 to entry: In orthonormal Cartesian coordinates, the three coordinates of the gradient are: $\frac{\partial \text{\hspace{0.17em}}f}{\partial \text{\hspace{0.17em}}x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{\hspace{0.17em}}f}{\partial \text{\hspace{0.17em}}y}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{\hspace{0.17em}}f}{\partial \text{\hspace{0.17em}}z}$. Note 3 to entry: The gradient of the scalar field | ||

Publication date: | 2008-08 | ||

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Internal notes: | 2017-02-20: Editorial revisions in accordance with the information provided in C00019 (IEV 102) - evaluation. JGO 2017-08-24: Order of <i> and <b> tags corrected. LMO | ||

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Note 1 to entry: The gradient expresses the variation of the scalar field quantity from the given point to a point at an infinitesimal distance $\mathrm{d}s$ in the direction of a given unit vector ** e** by the scalar product d

Note 2 to entry: In orthonormal Cartesian coordinates, the three coordinates of the gradient are:

$\frac{\partial \text{\hspace{0.17em}}f}{\partial \text{\hspace{0.17em}}x}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{\hspace{0.17em}}f}{\partial \text{\hspace{0.17em}}y}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\partial \text{\hspace{0.17em}}f}{\partial \text{\hspace{0.17em}}z}$.

Note 3 to entry: The gradient of the scalar field *f* is denoted **grad** *f* or $\mathbf{\nabla}\text{\hspace{0.05em}}f$.