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Assume that (x, y, z) is a given Cartesian coordinate system, that A is a vector field with components = (,,), and that = (,,) is a scalar field. The differential operator introduced by William Rowan Hamilton , written ∇ and called del or nabla, is symbolically defined in the form of a vector,
It can be thought of as the rate of change of the function in the -direction.. Sometimes, for = (,, …), the partial derivative of with respect to is denoted as . Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:
The differential was first introduced via an intuitive or heuristic definition by Isaac Newton and furthered by Gottfried Leibniz, who thought of the differential dy as an infinitely small (or infinitesimal) change in the value y of the function, corresponding to an infinitely small change dx in the function's argument x.
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
D: divergence, C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do ...
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Gottfried Wilhelm von Leibniz (1646–1716), German philosopher, mathematician, and namesake of this widely used mathematical notation in calculus.. In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinitesimal) increments of x and y, respectively ...
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