Nabla

Nabla , noted $\backslash \; nabla$, is a mathematical symbol being able as well to indicate the Gradient of a function in Analyze as a Connection of Koszul in differential Géométrie. The two concepts are obviously connected, which explains the use of the same symbol. In physics, it is used in an abstract way in dimension 3 to easily represent the divergence (), the Rotationnel () and the vectorial Laplacian () of a vector Field, as well as the Gradient () and the Laplacien () of a scalar Champ. These concepts are fundamental in physics, in particular in electromagnetism and Hydrodynamique.

Historical origin

The form of Nabla comes from a Delta ($\backslash \; Delta$) reversed, because of a comparable use (differential calculus), it was introduced by Peter Guthrie TAIT in 1867. Initially called with mischievousness “atled” (delta with back) by James Maxwell, the name Nabla was given to him by TAIT on the opinion of William Robertson Smith, in 1870, by analogy of form with a Greek Harpe which in the Antiquité bore this name.

Use in vectorial analysis

This is a list of some formulas of vectorial Analyze of general employment while working with several frames of reference.

\ overrightarrow \ nabla (\ overrightarrow \ nabla \ cdot \ overrightarrow {has}) - \ overrightarrow \ nabla^2 \ overrightarrow {has} \ overrightarrow {\ operatorname {grad}} \ \ operatorname {div} \ overrightarrow has - \ Delta \ overrightarrow A

$\backslash \; Delta\; F\; G\; =\; F\; \backslash \; Delta\; G\; +\; 2\; \backslash \; overrightarrow\; \backslash \; nabla\; F\; \backslash \; cdot\; \backslash \; overrightarrow\; \backslash \; nabla\; G\; +\; G\; \backslash \; Delta\; f$

Formule of Lagrange for the vector Product:
$\backslash \; overrightarrow\; \{has\}\; \backslash \; wedge\; (\backslash \; overrightarrow\; \{B\}\; \backslash \; wedge\; \backslash \; overrightarrow\; \{C\})$

\ overrightarrow {B} (\ overrightarrow {has} \ cdot \ overrightarrow {C}) - \ overrightarrow {C} (\ overrightarrow {has} \ cdot \ overrightarrow {B})

|- |+ Table with the $\backslash \; nabla$ (Nabla or LED) in the cylindrical or spherical coordinates |}

• Note: the spherical coordinates would have been more natural if $\backslash \; theta$ had been defined like the angle with plan X there.

Attention with the use of these operators: it is not a question of scalar products, but well of applications, in spite of the notation $\backslash \; overrightarrow\; \backslash \; nabla\; \backslash \; cdot\; \backslash \; overrightarrow\; A$. The result is the same one for the Cartesian coordinates, but becomes false for the curvilinear coordinates.

See too

Internal bonds

• vectorial Analysis
• Frame of reference (coordinated curvilinear)
• Coordinated polar
• Pseudovecteur

External bonds

• Of the name of Nabla
• First use of the symbol
• Genesis of the name: a whole history

Random links:

Series of the standards ISO 9000 | Conference of Dumbarton Oaks | Jacob Jordaens | Counter-gambit Albin | Widmanstaetten | Henri Dumat | Paliuli

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