- Motivation
- Notation
- Definition
- Gradient and The Derivative Or Differential
- Further Properties and Applications
- Generalizations
- See Also
- References
- Further Reading
- External Links

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), independent of 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, moving away from (x, y, z). 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 plane vector pointing in the direction of the steepest slope or gradeat 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%. A road going directly uphill has slope 40%, but a road going around the hill at an angle will have...The gradient of a function f {\\displaystyle f} at point a {\\displaystyle a} is usually written as ∇ f ( a ) {\\displaystyle \ abla f(a)} . It may also be denoted by any of the following: 1. ∇ → f ( a ) {\\displaystyle {\\vec {\ abla }}f(a)} : to emphasize the vector nature of the result. 2. grad f 3. ∂ f ∂ x | x = a {\\displaystyle \\left.{\\frac {\\partial f}{\\partial x}}\\right|_{x=a}} 4. ∂ i f {\\displaystyle \\partial _{i}f} and f i {\\displaystyle f_{i}} : Einstein notation.

The gradient (or gradient vector field) of a scalar function f(x1, x2, x3, …, xn) 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 vector v at each point x is the directional derivative of f along v. That is, 1. ( ∇ f ( x ) ) ⋅ v = D v f ( x ) . {\\displaystyle {\\big (}\ abla f(x){\\big )}\\cdot \\mathbf {v} =D_{\\mathbf {v} }f(x).} Formally, the gradient is dual to the derivative; see relationship with derivative. 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). The magnitude and direction of the gradient vector are independent of the particular coordinate representation.

The gradient is closely related to the (total) derivative ((total) differential) d f {\\displaystyle df} : they are transpose (dual) to each other. Using the convention that vectors in R n {\\displaystyle \\mathbb {R} ^{n}} are represented by column vectors, and that covectors (linear maps R n → R {\\displaystyle \\mathbb {R} ^{n}\\to \\mathbb {R} } ) are represented by row vectors,[a] the gradient ∇ f {\\displaystyle \ abla f} and the derivative d f {\\displaystyle df} are expressed as a column and row vector, respectively, with the same components, but transpose of each other: 1. ∇ f ( p ) = [ ∂ f ∂ x 1 ( p ) ⋮ ∂ f ∂ x n ( p ) ] ; {\\displaystyle \ abla f(p)={\\begin{bmatrix}{\\frac {\\partial f}{\\partial x_{1}}}(p)\\\\\\vdots \\\\{\\frac {\\partial f}{\\partial x_{n}}}(p)\\end{bmatrix}};} 2. d f p = [ ∂ f ∂ x 1 ( p ) ⋯ ∂ f ∂ x n ( p ) ] . {\\displaystyle df_{p}={\\begin{bmatrix}{\\frac {\\partial f}{\\partial x_{1}}}(p)&\\cdots &{\\frac {\\partial f}{\\partial x_{n}}}(p)\\end{bmatrix}}.} While these both have t...

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 (∇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 Fis then normal to the sur...

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 integralalong 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.

Jacobian

The Jacobian matrix 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. Suppose f : ℝn → ℝm is a function such that each of its first-order partial derivatives exist on ℝn. Then the Jacobian matrix of f is defined to be an m×n matrix, denoted by J f ( x ) {\\displaystyle \\mathbf {J} _{\\mathbb...

Gradient of a vector field

Since the total derivative of a vector field is a linear mapping from vectors to vectors, it is a tensorquantity. In rectangular coordinates, the gradient of a vector field f = ( f1, f2, f3)is defined by: 1. ∇ f = g j k ∂ f i ∂ x j e i ⊗ e k , {\\displaystyle \ abla \\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 ei and ek is a dyadic tensorof type (2,0))....

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, 1. g ( ∇ f , X ) = ∂ X f , {\\displaystyle g(\ abla f,X)=\\partial _{X}f,} that is, 1. g x ( ( ∇ f ) x , X x ) = ( ∂ X f ) ( x ) , {\\displaystyle g_{x}{\\big (}(\ abla f)_{x},X_{x}{\\big )}=(\\partial _{X}f)(x),} where gx( , ) 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 directio...

Bachman, David (2007), Advanced Calculus Demystified, New York: McGraw-Hill, ISBN 978-0-07-148121-2Beauregard, 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-XDowning, Douglas, Ph.D. (2010), Barron's E-Z Calculus, New York: Barron's, ISBN 978-0-7641-4461-5Dubrovin, 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...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. ISB...

"Gradient". Khan Academy.Kuptsov, L.P. (2001) [1994], "Gradient", Encyclopedia of Mathematics, EMS Press.Weisstein, Eric W. "Gradient". MathWorld.#### 花蓮+1！女大生被工頭爸傳染足跡

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- February 10, 1961; 60 years ago
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- Crystal Structure
- Crystal Faces and Shapes
- Occurrence in Nature
- Polymorphism and Allotropy
- Crystallization
- Defects, Impurities, and Twinning
- Chemical Bonds
- Quasicrystals
- Special Properties from Anisotropy
- Crystallography

The scientific definition of a "crystal" is

**based on the microscopic arrangement of atoms inside it**,**called the crystal structure.**A crystal is a solid where the atoms form a periodic arrangement. (Quasicrystals are an exception, see below). Not all solids are crystals. For example, when liquid water starts freezing, the phase change begins with small ice crystals that grow until they fuse, forming a polycrystalline structure. In the final block of ice, each of the small crystals (called "crystallites" or "grains") is a true crystal with a periodic arrangement of atoms, but the whole polycrystal does not have a periodic arrangement of atoms, because the periodic pattern is broken at the grain boundaries. Most macroscopic inorganic solids are polycrystalline, including almost all metals, ceramics, ice, rocks, etc. Solids that are neither crystalline nor polycrystalline, such as glass, are called amorphous solids, also called glassy, vitreous, or noncrystalline. These have no periodic...Crystals are commonly recognized by their shape, consisting of flat faces with sharp angles. These shape characteristics are not necessaryfor a crystal—a crystal is scientifically defined by its microscopic atomic arrangement, not its macroscopic shape—but the characteristic macroscopic shape is often present and easy to see. Euhedral crystals are those with obvious, well-formed flat faces. Anhedralcrystals do not, usually because the crystal is one grain in a polycrystalline solid. The flat faces (also called facets) of a euhedral crystal are oriented in a specific way relative to the underlying atomic arrangement of the crystal: they are planes of relatively low Miller index. This occurs because some surface orientations are more stable than others (lower surface energy). As a crystal grows, new atoms attach easily to the rougher and less stable parts of the surface, but less easily to the flat, stable surfaces. Therefore, the flat surfaces tend to grow larger and smoother, until...

Rocks

By volume and weight, the largest concentrations of crystals in the Earth are part of its solid bedrock. Crystals found in rocks typically range in size from a fraction of a millimetre to several centimetres across, although exceptionally large crystals are occasionally found. As of 1999[update], the world's largest known naturally occurring crystal is a crystal of beryl from Malakialina, Madagascar, 18 m (59 ft) long and 3.5 m (11 ft) in diameter, and weighing 380,000 kg (840,000 lb). Some c...

Ice

Water-based ice in the form of snow, sea ice, and glaciers are common crystalline/polycrystalline structures on Earth and other planets. A single snowflake is a single crystal or a collection of crystals, while an ice cube is a polycrystal.

Organigenic crystals

Many living organisms are able to produce crystals, for example calcite and aragonite in the case of most molluscs or hydroxylapatite in the case of vertebrates.

The same group of atoms can often solidify in many different ways. Polymorphism is the ability of a solid to exist in more than one crystal form. For example, water ice is ordinarily found in the hexagonal form Ice Ih, but can also exist as the cubic Ice Ic, the rhombohedral ice II, and many other forms. The different polymorphs are usually called different phases. In addition, the same atoms may be able to form noncrystalline phases. For example, water can also form amorphous ice, while SiO2 can form both fused silica (an amorphous glass) and quartz(a crystal). Likewise, if a substance can form crystals, it can also form polycrystals. For pure chemical elements, polymorphism is known as allotropy. For example, diamond and graphite are two crystalline forms of carbon, while amorphous carbonis a noncrystalline form. Polymorphs, despite having the same atoms, may have wildly different properties. For example, diamond is among the hardest substances known, while graphite is so soft tha...

Crystallization is the process of forming a crystalline structure from a fluid or from materials dissolved in a fluid. (More rarely, crystals may be deposited directly from gas; see thin-film deposition and epitaxy.) Crystallization is a complex and extensively-studied field, because depending on the conditions, a single fluid can solidify into many different possible forms. It can form a single crystal, perhaps with various possible phases, stoichiometries, impurities, defects, and habits. Or, it can form a polycrystal, with various possibilities for the size, arrangement, orientation, and phase of its grains. The final form of the solid is determined by the conditions under which the fluid is being solidified, such as the chemistry of the fluid, the ambient pressure, the temperature, and the speed with which all these parameters are changing. Specific industrial techniques to produce large single crystals (called boules) include the Czochralski process and the Bridgman technique....

An ideal crystal has every atom in a perfect, exactly repeating pattern. However, in reality, most crystalline materials have a variety of crystallographic defects, places where the crystal's pattern is interrupted. The types and structures of these defects may have a profound effect on the properties of the materials. A few examples of crystallographic defects include vacancy defects (an empty space where an atom should fit), interstitial defects (an extra atom squeezed in where it does not fit), and dislocations (see figure at right). Dislocations are especially important in materials science, because they help determine the mechanical strength of materials. Another common type of crystallographic defect is an impurity, meaning that the "wrong" type of atom is present in a crystal. For example, a perfect crystal of diamond would only contain carbon atoms, but a real crystal might perhaps contain a few boron atoms as well. These boron impurities change the diamond's color to slight...

In general, solids can be held together by various types of chemical bonds, such as metallic bonds, ionic bonds, covalent bonds, van der Waals bonds, and others. None of these are necessarily crystalline or non-crystalline. However, there are some general trends as follows. Metals are almost always polycrystalline, though there are exceptions like amorphous metal and single-crystal metals. The latter are grown synthetically. (A microscopically-small piece of metal may naturally form into a single crystal, but larger pieces generally do not.) Ionic compound materials are usually crystalline or polycrystalline. In practice, large salt crystals can be created by solidification of a molten fluid, or by crystallization out of a solution. Covalently bonded solids (sometimes called covalent network solids) are also very common, notable examples being diamond and quartz. Weak van der Waals forces also help hold together certain crystals, such as crystalline molecular solids, as well as the...

A quasicrystal consists of arrays of atoms that are ordered but not strictly periodic. They have many attributes in common with ordinary crystals, such as displaying a discrete pattern in x-ray diffraction, and the ability to form shapes with smooth, flat faces. Quasicrystals are most famous for their ability to show five-fold symmetry, which is impossible for an ordinary periodic crystal (see crystallographic restriction theorem). The International Union of Crystallography has redefined the term "crystal" to include both ordinary periodic crystals and quasicrystals ("any solid having an essentially discrete diffraction diagram"). Quasicrystals, first discovered in 1982, are quite rare in practice. Only about 100 solids are known to form quasicrystals, compared to about 400,000 periodic crystals known in 2004. The 2011 Nobel Prize in Chemistry was awarded to Dan Shechtmanfor the discovery of quasicrystals.

Crystals can have certain special electrical, optical, and mechanical properties that glass and polycrystals normally cannot. These properties are related to the anisotropy of the crystal, i.e. the lack of rotational symmetry in its atomic arrangement. One such property is the piezoelectric effect, where a voltage across the crystal can shrink or stretch it. Another is birefringence, where a double image appears when looking through a crystal. Moreover, various properties of a crystal, including electrical conductivity, electrical permittivity, and Young's modulus, may be different in different directions in a crystal. For example, graphitecrystals consist of a stack of sheets, and although each individual sheet is mechanically very strong, the sheets are rather loosely bound to each other. Therefore, the mechanical strength of the material is quite different depending on the direction of stress. Not all crystals have all of these properties. Conversely, these properties are not qui...

Crystallography is the science of measuring the crystal structure (in other words, the atomic arrangement) of a crystal. One widely used crystallography technique is X-ray diffraction. Large numbers of known crystal structures are stored in crystallographic databases.

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