Search results
Results From The WOW.Com Content Network
In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. [citation needed]The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction ...
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 azimuthal angle is denoted by φ ∈ [ 0 , 2 π ] {\displaystyle \varphi \in [0,2\pi ]} : it is the angle between the x -axis and the projection of the radial vector onto the xy -plane.
In vector calculus the derivative of a vector y with respect to a scalar x is known as the tangent vector of the vector y, . Notice here that y : R 1 → R m . Example Simple examples of this include the velocity vector in Euclidean space , which is the tangent vector of the position vector (considered as a function of time).
The derivative with respect to a vector as discussed above can be generalized to a derivative with respect to a general multivector, called the multivector derivative. Let F {\displaystyle F} be a multivector-valued function of a multivector.
Another two-point formula is to compute the slope of a nearby secant line through the points (x − h, f(x − h)) and (x + h, f(x + h)). The slope of this line is (+) (). This formula is known as the symmetric difference quotient.
The original theorem given by J. M. Danskin in his 1967 monograph [1] provides a formula for the directional derivative of the maximum of a (not necessarily convex) directionally differentiable function. An extension to more general conditions was proven 1971 by Dimitri Bertsekas.
Hence the formula () is regarded as the directional derivative at point in the direction of . This is analogous to vector calculus, where the inner product of a vector v {\displaystyle v} with the gradient gives the directional derivative in the direction of v {\displaystyle v} .
Thus we have a formula for ∂ v f, (one of ways to represent the directional derivative) where v is arbitrary; for ():= [,] (see its full definition above), its directional derivative with respect to v is = = = where the first two equalities just show different representations of the directional derivative.