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In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change.
More rigorously, the gradient of a function from the Euclidean space Rn to R is the best linear approximation to that function at any particular point in Rn. To that extent, the gradient is a particular case of the Jacobian.
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The gradient is noted by:
where (nabla) is the vector differential operator del, and φ is a scalar function. It is sometimes also written grad(φ).
In 3 dimensions, the expression expands to
in Cartesian coordinates. (See partial derivative and vector.)
For example, the gradient of the function φ = 2x + 3y2 - sin(z) is:
For any differentiable function f on a manifold M, the gradient of f is the vector field such that for any vector ξ,
where ξf is the function that takes any point p to the directional derivative of f in the direction ξ evaluated at p. In other words, under some coordinate chart, ξf(p) will be: