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Table with the del operator in cartesian, cylindrical and spherical coordinates Operation Cartesian coordinates (x, y, z) Cylindrical coordinates (ρ, φ, z) Spherical coordinates (r, θ, φ), where θ is the polar angle and φ is the azimuthal angle α; Vector field A
Del operator, represented by the nabla symbol. Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus.
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
Due to linearity, vectors can be defined in any number of dimensions, as each component of the vector acts on the function separately. One mathematical example is the del operator, which is itself a vector (useful in momentum-related quantum operators, in the table below). An operator in n-dimensional space can be written:
Del squared may refer to: Laplace operator, a differential operator often denoted by the symbol ∇ 2; Hessian matrix, sometimes denoted by ∇ 2; Aitken's delta-squared process, a numerical analysis technique used for accelerating the rate of convergence of a sequence
basis vectors that are locally normal to the isosurface created by the other coordinates: = are covariant vectors (denoted by raised indices), ∇ is the del operator. Note that, because of Einstein's summation convention, the position of the indices of the vectors is the opposite of that of the coordinates.
Del in cylindrical and spherical coordinates – Mathematical gradient operator in certain coordinate systems; Double Fourier sphere method; Elevation (ballistics) – Angle in ballistics; Euler angles – Description of the orientation of a rigid body; Gimbal lock – Loss of one degree of freedom in a three-dimensional, three-gimbal mechanism
It is written as = or = or = where , which is the fourth power of the del operator and the square of the Laplacian operator (or ), is known as the biharmonic operator or the bilaplacian operator.