NationMaster - Encyclopedia: Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P, in the direction of V. It therefore generalizes the notion of a partial derivative, in which the direction is always taken parallel to one of the coordinate axes. Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... A vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a magnitude and a direction. The word vector is also now used for more general concepts (see also vector and generalizations below), but in this... 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 constant (as opposed to the total derivative, in which all variables are allowed to vary). ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

Definition

The directional derivative of a scalar function $f(vec{x}) = f(x_1, x_2, ldots, x_n)$ along a vector $vec{v} = (v_1, ldots, v_n)$ is the function defined by the limit In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... Partial plot of a function f. ... In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...

if $vec{v}$ is the unit vector, $vec{v} = langle a,b rangle$ , then the directional derivitive is

And if the function is a differentiable function of

x

and

y

then the function has a derivative in the direction of any unit vector $vec{u} = langle a,b rangle$ and

If the function is differentiable, it can be written in terms of the gradient $nabla(f)$ of

f

by Horizontal line (use sparingly)d grade for the grade or gradient of roads and other geographic features. ...

where $cdot$ denotes the dot product (Euclidean inner product). At any point

p

, the directional derivative of

f

intuitively represents the rate of change in

f

along $vec{v}$ at the point

p

. Usually directions are taken to be normalized, so $vec{v}$ is a unit vector, although the definition above works for arbitrary (even zero) vectors. In mathematics, the dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. ... In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, a derivative is the rate of change of a quantity (e. ... In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1. ...

Proof of theorem 1

The directional derivative in differential geometry

A vector field at a point

p

naturally gives rise to linear functionals defined on

p

by evaluating the directional derivative of a differentiable function

f

along the vector $vec{v}$ where $vec{v}$ is the vector of the tangent space at

p

assigned by the vector field. The value of the functional is then defined as the value of the corresponding directional derivative at

p

in the direction of $vec{v}$ . Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

Normal derivative

A normal derivative is a directional derivative taken in the direction normal (that is, orthogonal) to some surface in space, or more generally along a normal vector field orthogonal to some hypersurface. See for example Neumann boundary condition. In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... A normal vector is a vector which is perpendicular to a surface or manifold. ... In mathematics, a hypersurface is some kind of submanifold. ... In mathematics, a Neumann boundary condition imposed on an ordinary differential equation or a partial differential equation specifies the values the derivative of a solution is to take on the boundary of the domain. ...

Contents

Definition

Proof of theorem 1

The directional derivative in differential geometry

Normal derivative

See also

Contents

Definition GA_googleFillSlot("encyclopedia_square");

Proof of theorem 1

The directional derivative in differential geometry

Normal derivative

See also

Definition