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A contour plot of (,) = +, showing the gradient vector in black, and the unit vector scaled by the directional derivative in the direction of in orange. The gradient vector is longer because the gradient points in the direction of greatest rate of increase of a function.
English: This plot shows the gradient, ^, and ^ scaled by the directional derivative in the direction of ^ of (,) = + at (−5, 5). Created with the following Sage code, and later edited in Inkscape:
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 ...
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
The second derivative test can still be used to analyse critical points by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum.
where the right-hand side is the directional derivative and there are many ways to represent it. Formally, the derivative is dual to the gradient; see relationship with derivative. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (see Spatial gradient).
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This allows the abstraction of the notion of a directional derivative of a scalar function to general manifolds. For manifolds that are subsets of R n, this tangent vector will agree with the directional derivative. The differential or pushforward of a map between manifolds is the induced map between tangent spaces of those maps.