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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 gradient is related to the differential by the formula = for any , where is the dot product: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector.
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 ...
These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations. [1] The directional derivative provides a systematic way of finding these derivatives. [2]
From this follows that the directional derivative is the inner product of its direction by the vector derivative. All needs to be observed is that the direction a {\displaystyle a} can be written a = ( a ⋅ e i ) e i {\displaystyle a=(a\cdot e^{i})e_{i}} , so that:
Consequently, the gradient determines a vector field. [42] Directional derivatives ... These are measured using directional derivatives. Given a vector ...
The directional derivative of a scalar function f(x) of the space vector x in the direction of the unit vector u (represented in this case as a column vector) is defined using the gradient as follows.
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.