Gradient

This article is about the gradient of a multivariate function. For colour gradients, see colour gradient. For direction and steepness of a line on a graph, see Slope. For the slope of a road or other physical feature, see Grade (slope). For the similarly spelled unit of angle also known as gon, see Gradian. For other uses, see Gradient (disambiguation).

The gradient, represented by the blue arrows, denote the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).

In vector calculus, the gradient is a multi-variable generalization of the derivative. Whereas the ordinary derivative of a function of a single variable is a scalar-valued function, the gradient of a function of several variables is a vector-valued function. Specifically, the gradient of a differentiable function $f$ of several variables, at a point $P$, is the vector whose components are the partial derivatives of $f$ at $P$.

Much as the derivative of a function of a single variable represents the slope of the tangent to the graph of the function, if at a point $P$, the gradient of a function of several variables is not the zero vector, it has the direction of fastest increase of the function at $P$, and its magnitude is the rate of increase in that direction.

The magnitude and direction of the gradient vector are independent of the particular coordinate representation.

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

Gradient of the 2D function f(x, y) = xe^{−(x2 + y2)} is plotted as blue arrows over the pseudocolor plot of the function.

Consider a room where 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 of the function f(x,y) = −(cos²x + cos²y)² depicted as a projected vector field on the bottom plane.

The gradient (or gradient vector field) of a scalar function f(x₁, x₂, x₃, ..., x_n) is denoted ∇f or ∇→f where ∇ (nabla) denotes the vector differential operator, del. The notation grad f is also commonly used to represent the gradient. The gradient of f is defined as the unique vector field whose dot product with any unit vector v at each point x is the directional derivative of f along v. That is,

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

When a function also depends on a parameter such as time, the gradient often refers simply to the vector of its spatial derivatives only (see Spatial gradient).

Cartesian coordinates

In the three-dimensional Cartesian coordinate system with a Euclidean metric, the gradient, if it exists, 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 in the directions of the x, y and z coordinates, respectively. 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

Main article: Del in cylindrical and spherical coordinates

In cylindrical coordinates with a Euclidean metric, the gradient is given by:

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

where ρ is the axial distance, φ is the azimuthal or azimuth 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:

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

where r is the radial distance, φ is the azimuthal angle and θ is the polar angle, and e_r, e_θ and e_φ are again local unit vectors pointing in the coordinate directions (i.e. the normalized covariant basis).

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

General coordinates

We consider general coordinates, which we write as x¹, ..., xⁱ, ..., xⁿ, where n is the number of dimensions of the domain. Here, the upper index refers to the position in the list of the coordinate or component, so x² refers to the second component—not the quantity x squared. The index variable i refers to an arbitrary element xⁱ. Using Einstein notation, the gradient can then be written as:

$\nabla f={\frac {\partial f}{\partial x^{i}}}g^{ij}\mathbf {e} _{j}$ ( Note that its dual is $\mathrm {d} f={\frac {\partial f}{\partial x^{i}}}\mathbf {e} ^{i}$ ),

where $\mathbf {e} _{i}=\partial \mathbf {x} /\partial x^{i}$ and $\mathbf {e} ^{i}=\mathrm {d} x^{i}$ refer to the unnormalized local covariant and contravariant bases respectively, $g^{ij}$ is the inverse metric tensor, and the Einstein summation convention implies summation over i and j.

If the coordinates are orthogonal we can easily express the gradient (and the differential) in terms of the normalized bases, which we refer to as ${\hat {\mathbf {e} }}_{i}$ and ${\hat {\mathbf {e} }}^{i}$, using the scale factors (also known as Lamé coefficients) $h_{i}=\lVert \mathbf {e} _{i}\rVert =1\,/\lVert \mathbf {e} ^{i}\,\rVert$ :

$\nabla f=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} _{i}$ ( and $\mathrm {d} f=\sum _{i=1}^{n}\,{\frac {\partial f}{\partial x^{i}}}{\frac {1}{h_{i}}}\mathbf {\hat {e}} ^{i}$ ),

where we cannot use Einstein notation, since it is impossible to avoid the repetition of more than two indices. Despite the use of upper and lower indices, $\mathbf {\hat {e}} _{i}$, $\mathbf {\hat {e}} ^{i}$, and $h_{i}$ are neither contravariant nor covariant.

The latter expression evaluates to the expressions given above for cylindrical and spherical coordinates.

Gradient and the derivative or differential

Linear approximation to a function

The gradient of a function f from the Euclidean space Rⁿ to R at any particular point x₀ in Rⁿ characterizes the best linear approximation to f at x₀. The approximation is as follows:

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

for x close to x₀, where (∇f )_x0 is the gradient of f computed at x₀, and the dot denotes the dot product on Rⁿ. This equation is equivalent to the first two terms in the multivariable Taylor series expansion of f at x₀.

Differential or (exterior) derivative

The best linear approximation to a differentiable function

$f\colon \mathbf {R} ^{n}\to \mathbf {R}$

at a point x in Rⁿ is a linear map from Rⁿ to R which is often denoted by df_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

$(\nabla f)_{x}\cdot v=df_{x}(v)$

for any v ∈ Rⁿ. The function df, which maps x to df_x, is called the differential or exterior derivative of f and is an example of a differential 1-form.

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

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

so that df_x(v) is given by matrix multiplication. Assuming the standard Euclidean metric on Rⁿ, the gradient is then the corresponding column vector, i.e.,

$(\nabla f)_{i}=df_{i}^{\mathsf {T}}.$

Gradient as a derivative

Let U be an open set in Rⁿ. If the function f : U → R is differentiable, then the differential of f is the (Fréchet) derivative of f. Thus ∇f is a function from U to the space Rⁿ 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 ∈ Rⁿ, 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 ∈ Rⁿ, 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 → R is a real-valued function defined on a subset A of Rⁿ, 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 → Rⁿ maps a subset I ⊂ R into Rⁿ. If g is differentiable at a point c ∈ I such that g(c) = a, then

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

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

More generally, if instead I ⊂ 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 → R is a real valued function on a subset I of R, and that h is differentiable at the point f(a) ∈ I. Then

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

Further properties and applications

Level sets

See also: Level set § Level sets versus the gradient

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 (∇f )_x ⋅ 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 dF 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(x₁, ..., 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

Main article: 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.

Generalizations

Gradient of a vector

See also: Covariant derivative

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 = ( f¹, f², f³) is defined by:

$\nabla \mathbf {f} =g^{jk}{\frac {\partial f^{i}}{\partial x^{j}}}\mathbf {e} _{i}\otimes \mathbf {e} _{k},$

(where the Einstein summation notation is used and the tensor product of the vectors e_i and e_k is a dyadic tensor of type (2,0)). Overall, this expression equals the transpose of 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:

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

where g^jk are the components of the inverse metric tensor and the e_i are the coordinate basis vectors.

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

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

where ∇_c is the connection.

Riemannian manifolds

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

$g(\nabla f,X)=\partial _{X}f,$

i.e.,

$g_{x}{\big (}(\nabla f)_{x},X_{x}{\big )}=(\partial _{X}f)(x),$

where g_x( , ) denotes the inner product of tangent vectors at x defined by the metric g and ∂_X f is the function that takes any point x ∈ 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 Rⁿ, (∂_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 jth component of X in this coordinate chart.

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

$\nabla f=g^{ik}{\frac {\partial f}{\partial x^{k}}}{\textbf {e}}_{i}.$

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

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

More precisely, the gradient ∇f is the vector field associated to the differential 1-form df 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 Rⁿ is a special case of this in which the metric is the flat metric given by the dot product.

See also

• Curl
• Divergence
• Four-gradient
• Hessian matrix
• Skew gradient

References

Sources

• Bachman, David (2007), Advanced Calculus Demystified, New York: McGraw-Hill, ISBN 978-0-07-148121-2
• Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Company, ISBN 0-395-14017-X
• Downing, Douglas, Ph.D. (2010), Barron's E-Z Calculus, New York: Barron's, ISBN 978-0-7641-4461-5{{citation}}: CS1 maint: multiple names: authors list (link)
• 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)
• Harper, Charlie (1976), Introduction to Mathematical Physics, New Jersey: Prentice-Hall, ISBN 0-13-487538-9
• Kreyszig, Erwin (1972), Advanced Engineering Mathematics (3rd ed.), New York, NY: Wiley, ISBN 0-471-50728-8
• "McGraw Hill Encyclopedia of Science & Technology". McGraw-Hill Encyclopedia of Science & Technology (10th ed.). New York: McGraw-Hill. 2007. ISBN 978-0-07-144143-8.
• Moise, Edwin E. (1967), Calculus: Complete, Reading: Addison-Wesley
• Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042
• 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)
• Stoker, J. J. (1969), Differential Geometry, New York: Wiley, ISBN 0-471-82825-4
• Swokowski, Earl W.; Olinick, Michael; Pence, Dennis; Cole, Jeffery A. (1994), Calculus (6th ed.), Boston: PWS Publishing Company, ISBN 0-534-93624-5

Further reading

• 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)

External links

• "Gradient". Khan Academy.
• Kuptsov, L.P. (2001) [1994], "Gradient", Encyclopedia of Mathematics, EMS Press.
• Weisstein, Eric W. "Gradient". MathWorld.