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Encyclopedia > Normal derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector intuitively represents the rate of change of the function in the direction of that vector. It therefore generalizes the notion of a partial derivative, in which the direction is always taken parallel to one of the coordinate axes. Euclid, detail from The School of Athens by Raphael. ... In mathematics, the derivative of a function is one of the two central concepts of calculus. ... The word vector means carrier in Latin; it is derived from the Latin verb vehere, which means to carry. ... 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. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ...

1 Definition
2 The directional derivative in differential geometry
3 Normal derivative
4 See also

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 The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ... 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 larger and larger; or the behavior of a sequences elements, as their index becomes larger and larger. ...

$D_{vec{v}}{f} = lim_{h rightarrow 0}{frac{f(vec{x} + hvec{v}) - f(vec{x})}{h}}.$

If the function is differentiable, it can be written in terms of the gradient $nabla(f)$ of $f$ by In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

$D_{vec{v}}{f} = nabla(f) cdot vec{v}$

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 $f$ in the direction of $vec{v}$ at the point $p$ . 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. ... // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity Î± resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ... In mathematics, the derivative is one of the two central concepts of calculus. ...

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 needs to be introduced when generalizing 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. ...

Categories: Multivariate calculus | Differential geometry In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ... 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. ... In mathematics, a Lie derivative, named after Sophus Lie, is a derivation on the algebra of smooth functions over a manifold M. The vector space of all Lie derivatives on M forms an infinite dimensional Lie algebra with respect to the Lie bracket defined by The Lie derivatives are represented... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the GÃ¢teaux derivative is a generalisation in functional analysis of the standard derivative of the differential calculus. ...

Results from FactBites:

Covariant derivative - Definition, explanation (1108 words)
	In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
	Here we give a traditional index-notation introduction to the covariant derivative (also known as the tensor derivative) of a vector with respect to a vector field; the covariant derivative of a tensor is an extension of the same concept.
	Often a notation is used in which the covariant derivative is given with a semicolon, while a normal derivative is indicated by a comma.

Covariant derivative - Wikipedia, the free encyclopedia (2317 words)
	Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach via a connection form.
	The covariant derivative of a tensor field is presented as an extension of the same concept.
	Often a notation is used in which the covariant derivative is given with a semicolon, while a normal partial derivative is indicated by a comma.

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The directional derivative in differential geometry

Normal derivative

See also

Definition