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A directional derivative is a concept in multivariable calculus that 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 velocity specified by v.
The directional derivative of a scalar function f with respect to a vector v at a point (e.g., position) x may be denoted by any of the following:
It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
The directional derivative is a special case of the Gateaux derivative.
and 27 Related for: Directional derivative information
A directionalderivative is a concept in multivariable calculus that measures the rate at which a function changes in a particular direction at a given...
directionalderivatives. Choose a vector v = ( v 1 , … , v n ) {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})} , then the directionalderivative of...
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held...
Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directionalderivative onto the manifold's tangent...
is the rate of increase in that direction, the greatest absolute directionalderivative. Further, a point where the gradient is the zero vector is known...
{\displaystyle F} be a multivector-valued function of a vector. The directionalderivative of F {\displaystyle F} along b {\displaystyle b} at a {\displaystyle...
difference in the definition of the limit and differentiation. Directional limits and derivatives define the limit and differential along a 1D parametrized...
expressions for the gradient, divergence, curl, directionalderivative, and Laplacian. The vector derivative of a scalar field f {\displaystyle f} is called...
derivative of a tensor field with respect to a vector field would be to take the components of the tensor field and take the directionalderivative of...
derivative of a function on a differentiable manifold, the most fundamental of which is the directionalderivative. The definition of the directional...
covariant derivative makes a choice for taking directionalderivatives of vector fields along curves. This extends the directionalderivative of scalar...
as directionalderivatives. Given a vector v {\displaystyle v} in R n {\displaystyle \mathbb {R} ^{n}} , one defines the corresponding directional derivative...
like the Gateaux derivative is preferred. In many practical cases, the functional differential is defined as the directionalderivative δ F [ ρ , ϕ ] =...
notation just defined for the derivative of a scalar with respect to a vector we can re-write the directionalderivative as ∇ u f = ∂ f ∂ x u . {\displaystyle...
every smooth vector field X, df (X) = dX f , where dX f is the directionalderivative of f in the direction of X. The exterior product of differential...
logarithmic derivative of a function f is defined by the formula f ′ f {\displaystyle {\frac {f'}{f}}} where f ′ {\displaystyle f'} is the derivative of f....
} where A ⋅ ∇ {\displaystyle \mathbf {A} \cdot \nabla } is the directionalderivative in the direction of A {\displaystyle \mathbf {A} } multiplied by...
function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same...
second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. Informally, the second derivative can be...
general case is to use the total derivative, which is a linear transformation that captures all directionalderivatives in a single formula. Consider differentiable...
applications, the directionalderivative is indeed sufficient. The above arithmetic can be generalized to calculate second order and higher derivatives of multivariate...
derivative of the field u·(∇y), or as involving the streamline directionalderivative of the field (u·∇) y, leading to the same result. Only this spatial...
mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directionalderivative in differential calculus. Named after René...
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h ( x ) = f (...
In mathematics, the Hadamard derivative is a concept of directionalderivative for maps between Banach spaces. It is particularly suited for applications...
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