Gradient

In the above two images, the values of the function are represented in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.

In mathematics, the gradient is a generalization of the usual concept of derivative to functions of several variables. If $f (x 1, ..., x n)$ is a differentiable, real-valued function of several variables, its gradient is the vector whose components are the $n$ partial derivatives of $f$ . It is thus a vector-valued function.

Similarly to the usual derivative, the gradient represents the slope of the tangent of the graph of the function. More precisely, the gradient points in the direction of the greatest rate of increase of the function, and its magnitude is the slope of the graph in that direction. The components of the gradient in coordinates are the coefficients of the variables in the equation of the tangent space to the graph. This characterizing property of the gradient allows it to be defined independently of a choice of coordinate system, as a vector field whose components in a coordinate system will transform when going from one coordinate system to another.

The Jacobian is the generalization of the gradient for vector-valued functions of several variables and differentiable maps between Euclidean spaces or, more generally, manifolds. A further generalization for a function between Banach spaces is the Fréchet derivative.

Motivation

Consider a room in which the temperature is given by a scalar field, $T$ , so at each point $(x, y, z)$ the temperature is $T (x, y, z)$ . (Assume that the temperature does not change over time.) At each point in the room, the gradient of $T$ at that point will show the direction in which the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.

Consider a surface whose height above sea level at point $(x, y)$ is $H (x, y)$ . The gradient of $H$ at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.

The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product. Suppose that the steepest slope on a hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%. If, instead, the road goes around the hill at an angle, then it will have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20%, which is 40% times the cosine of 60°.

This observation can be mathematically stated as follows. If the hill height function $H$ is differentiable, then the gradient of $H$ dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when $H$ is differentiable, the dot product of the gradient of $H$ with a given unit vector is equal to the directional derivative of $H$ in the direction of that unit vector.

Definition

The gradient (or gradient vector field) of a scalar function $f (x 1, x 2, x 3,... x n)$ is denoted $\nabla f$ or $\nabla \to f$ where $\nabla$ (the nabla symbol) denotes the vector differential operator, del. The notation $grad f$ is also commonly used for the gradient. The gradient of $f$ is defined as the unique vector field whose dot product with any vector $v$ at each point $x$ is the directional derivative of $f$ along $v$ . That is,

{\big (}\nabla f(x){\big )}\cdot \mathbf {v} =D_{\mathbf {v} }f(x).

In a rectangular coordinate system, the gradient is the vector field whose components are the partial derivatives of $f$ :

\nabla f={\frac {\partial f}{\partial x_{1}}}\mathbf {e} _{1}+\cdots +{\frac {\partial f}{\partial x_{n}}}\mathbf {e} _{n}

where the $e i$ are the orthogonal unit vectors pointing in the coordinate directions. When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only.

Cartesian coordinates

In the three-dimensional Cartesian coordinate system, the gradient is given by:

\nabla f={\frac {\partial f}{\partial x}}\mathbf {i} +{\frac {\partial f}{\partial y}}\mathbf {j} +{\frac {\partial f}{\partial z}}\mathbf {k}

where $i$ , $j$ , $k$ are the standard unit vectors. For example, the gradient of the function

f(x,y,z)=2x+3y^{2}-\sin(z)

is

\nabla f=2\mathbf {i} +6y\mathbf {j} -\cos(z)\mathbf {k} .

In some applications it is customary to represent the gradient as a row vector or column vector of its components in a rectangular coordinate system.

Cylindrical and spherical coordinates

In cylindrical coordinates, the gradient is given by:^[1]

\nabla f(\rho ,\varphi ,z)={\frac {\partial f}{\partial \rho }}\mathbf {e} _{\rho }+{\frac {1}{\rho }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }+{\frac {\partial f}{\partial z}}\mathbf {e} _{z}

where $φ$ is the azimuthal or azmuth angle, $z$ is the axial coordinate, and $e ρ$ , $e φ$ and $e z$ are unit vectors pointing along the coordinate directions.

In spherical coordinates, the gradient is given by:^[1]

\nabla f(r,\theta ,\varphi )={\frac {\partial f}{\partial r}}\mathbf {e} _{r}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}\mathbf {e} _{\theta }+{\frac {1}{r\sin \theta }}{\frac {\partial f}{\partial \varphi }}\mathbf {e} _{\varphi }

where $φ$ is the azimuth angle and $θ$ is the zenith angle.

For the gradient in other orthogonal coordinate systems, see Orthogonal coordinates (Differential operators in three dimensions).

Gradient and the derivative or differential

Linear approximation to a function

The gradient of a function $f$ from the Euclidean space $ℝ n$ to $ℝ$ at any particular point $x 0$ in $ℝ n$ characterizes the best linear approximation to $f$ at $x 0$ . The approximation is as follows:

f(x)\approx f(x_{0})+(\nabla f)_{x_{0}}\cdot (x-x_{0})

for $x$ close to $x 0$ , where $(\nabla f) x 0$ is the gradient of $f$ computed at $x 0$ , and the dot denotes the dot product on $ℝ n$ . This equation is equivalent to the first two terms in the multivariable Taylor series expansion of $f$ at $x 0$ .

Differential or (exterior) derivative

The best linear approximation to a differentiable function

f:\mathbb {R} ^{n}\to \mathbb {R}

at a point $x$ in $ℝ n$ is a linear map from $ℝ n$ to $ℝ$ which is often denoted by $d f x$ or $Df (x)$ and called the differential or (total) derivative of $f$ at $x$ . The gradient is therefore related to the differential by the formula

(\nabla f)_{x}\cdot v=\mathrm {d} f_{x}(v)

for any $v \in ℝ n$ . The function $d f$ , which maps $x$ to $d f x$ , is called the differential or exterior derivative of $f$ and is an example of a differential 1-form.

If $ℝ n$ is viewed as the space of (dimension $n$ ) column vectors (of real numbers), then one can regard $d f$ as the row vector with components

\left({\frac {\partial f}{\partial x_{1}}},\dots ,{\frac {\partial f}{\partial x_{n}}}\right)

so that $d f x (v)$ is given by matrix multiplication. The gradient is then the corresponding column vector, i.e.,

(\nabla f)_{i}=\mathrm {d} f_{i}^{\mathsf {T}}

.

Gradient as a derivative

Let $U$ be an open set in $ℝ n$ . If the function $f : U \to ℝ$ is differentiable, then the differential of $f$ is the (Fréchet) derivative of $f$ . Thus $\nabla f$ is a function from $U$ to the space $ℝ$ such that

\lim _{h\to 0}{\frac {\|f(x+h)-f(x)-\nabla f(x)\cdot h\|}{\|h\|}}=0

where · is the dot product.

As a consequence, the usual properties of the derivative hold for the gradient:

Linearity

The gradient is linear in the sense that if $f$ and $g$ are two real-valued functions differentiable at the point $a \in ℝ n$ , and $α$ and $β$ are two constants, then $αf + βg$ is differentiable at $a$ , and moreover

\nabla \left(\alpha f+\beta g\right)(a)=\alpha \nabla f(a)+\beta \nabla g(a).

Product rule

If $f$ and $g$ are real-valued functions differentiable at a point $a \in ℝ n$ , then the product rule asserts that the product $fg$ is differentiable at $a$ , and

\nabla (fg)(a)=f(a)\nabla g(a)+g(a)\nabla f(a).

Chain rule

Suppose that $f : A \to ℝ$ is a real-valued function defined on a subset $A$ of $ℝ n$ , and that $f$ is differentiable at a point $a$ . There are two forms of the chain rule applying to the gradient. First, suppose that the function $g$ is a parametric curve; that is, a function $g : I \to ℝ n$ maps a subset $I \subset ℝ$ into $ℝ n$ . If $g$ is differentiable at a point $c \in I$ such that $g (c) = a$ , then

(f\circ g)'(c)=\nabla f(a)\cdot g'(c),

where ∘ is the composition operator: $(f \circ g)(x) = f (g (x))$ .

More generally, if instead $I \subset ℝ k$ , then the following holds:

\nabla (f\circ g)(c)={\big (}Dg(c){\big )}^{\mathsf {T}}{\big (}\nabla f(a){\big )}

where $(Dg)$ ^T denotes the transpose Jacobian matrix.

For the second form of the chain rule, suppose that $h : I \to ℝ$ is a real valued function on a subset $I$ of $ℝ$ , and that $h$ is differentiable at the point $f (a) \in I$ . Then

\nabla (h\circ f)(a)=h'{\big (}f(a){\big )}\nabla f(a).

Further properties and applications

Level sets

A level surface, or isosurface, is the set of all points where some function has a given value.

If $f$ is differentiable, then the dot product $(\nabla f) x \cdot v$ of the gradient at a point $x$ with a vector $v$ gives the directional derivative of $f$ at $x$ in the direction $v$ . It follows that in this case the gradient of $f$ is orthogonal to the level sets of $f$ . For example, a level surface in three-dimensional space is defined by an equation of the form $F (x, y, z) = c$ . The gradient of $F$ is then normal to the surface.

More generally, any embedded hypersurface in a Riemannian manifold can be cut out by an equation of the form $F (P) = 0$ such that $d F$ is nowhere zero. The gradient of $F$ is then normal to the hypersurface.

Similarly, an affine algebraic hypersurface may be defined by an equation $F (x 1,... x n) = 0$ , where $F$ is a polynomial. The gradient of $F$ is zero at a singular point of the hypersurface (this is the definition of a singular point). At a non-singular point, it is a nonzero normal vector.

Conservative vector fields and the gradient theorem

The gradient of a function is called a gradient field. A (continuous) gradient field is always a conservative vector field: its line integral along any path depends only on the endpoints of the path, and can be evaluated by the gradient theorem (the fundamental theorem of calculus for line integrals). Conversely, a (continuous) conservative vector field is always the gradient of a function.

Riemannian manifolds

For any smooth function $f$ on a Riemannian manifold $(M, g)$ , the gradient of $f$ is the vector field $\nabla f$ such that for any vector field X,

g(\nabla f,X)=\partial _{X}f,\qquad {\text{i.e.,}}\quad g_{x}{\big (}(\nabla f)_{x},X_{x}{\big )}=(\partial _{X}f)(x)

where $g x ( , )$ denotes the inner product of tangent vectors at $x$ defined by the metric $g$ and $\partial X f$ (sometimes denoted $X (f)$ ) is the function that takes any point $x \in M$ to the directional derivative of $f$ in the direction $X$ , evaluated at $x$ . In other words, in a coordinate chart $φ$ from an open subset of $M$ to an open subset of $ℝ n$ , $(\partial X f)(x)$ is given by:

\sum _{j=1}^{n}X^{j}{\big (}\varphi (x){\big )}{\frac {\partial }{\partial x_{j}}}(f\circ \varphi ^{-1}){\Bigg |}_{\varphi (x)}

where $X j$ denotes the $j$ th component of $X$ in this coordinate chart.

So, the local form of the gradient takes the form:

\nabla f=g^{ik}{\frac {\partial f}{\partial x^{k}}}{\frac {\partial }{\partial x^{i}}}.

Generalizing the case $M = ℝ n$ , the gradient of a function is related to its exterior derivative, since

(\partial _{X}f)(x)=(df)_{x}(X_{x}).

More precisely, the gradient $\nabla f$ is the vector field associated to the differential 1-form $d f$ using the musical isomorphism

\sharp =\sharp ^{g}\colon T^{*}M\to TM

(called "sharp") defined by the metric $g$ . The relation between the exterior derivative and the gradient of a function on $ℝ n$ is a special case of this in which the metric is the flat metric given by the dot product.

Generalizations

Gradient of a vector

Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensor quantity.

In rectangular coordinates, the gradient of a vector field $f = (f 1, f 2, f 3)$ is defined by:

\nabla \mathbf {f} =g^{jk}{\frac {\partial f^{i}}{\partial x_{j}}}\mathbf {e} _{i}\mathbf {e} _{k}

where the Einstein summation notation is used and the product of the vectors $e i$ , $e k$ is a dyadic tensor of type (2,0), or the Jacobian matrix

{\frac {\partial f_{i}}{\partial x_{j}}}={\frac {\partial (f_{1},f_{2},f_{3})}{\partial (x_{1},x_{2},x_{3})}}

.

In curvilinear coordinates, or more generally on a curved manifold, the gradient involves Christoffel symbols:

\nabla \mathbf {f} =g^{jk}\left({\frac {\partial f^{i}}{\partial x_{j}}}+{\Gamma ^{i}}_{jl}f^{l}\right)\mathbf {e} _{i}\mathbf {e} _{k}

where $g jk$ are the components of the metric tensor and the $e i$ are the coordinate vectors.

Expressed more invariantly, the gradient of a vector field $f$ can be defined by the Levi-Civita connection and metric tensor:^[2]

\nabla ^{a}\mathbf {f} ^{b}=g^{ac}\nabla _{c}\mathbf {f} ^{b}

where $\nabla c$ is the connection.

Notes

^ ^a ^b Schey 1992, pp. 139–142.
^ Dubrovin, Fomenko & Novikov 1991, pp. 348–349.

References

Korn, Theresa M.; Korn, Granino Arthur (2000). Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. Dover Publications. pp. 157–160. ISBN 0-486-41147-8. OCLC 43864234. {{cite book}}: Invalid |ref=harv (help)

Schey, H.M. (1992). Div, Grad, Curl, and All That (2nd ed.). W.W. Norton. ISBN 0-393-96251-2. OCLC 25048561. {{cite book}}: Invalid |ref=harv (help)

Dubrovin, B.A.; Fomenko, A.T.; Novikov, S.P. (1991). Modern Geometry--Methods and Applications: Part I: The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics (2nd ed.). Springer. ISBN 978-0-387-97663-1. {{cite book}}: Invalid |ref=harv (help)

External links

"Gradient". Khan Academy.

Kuptsov, L.P. (2001) [1994], "Gradient", Encyclopedia of Mathematics, EMS Press.

Weisstein, Eric W. "Gradient". MathWorld.

[Schey-1992-1] Schey 1992, pp. 139–142.

[2] Dubrovin, Fomenko & Novikov 1991, pp. 348–349.

[1]

[2]