From the school mathematics course we know that a vector on a plane is a directed segment. Its beginning and end have two coordinates. The vector coordinates are calculated by subtracting the start coordinates from the end coordinates.
The concept of a vector can be extended to n-dimensional space (instead of two coordinates there will be n coordinates).
Gradient gradzfunctionz=f(x 1, x 2, ...x n) is the vector of partial derivatives of the function at a point, i.e. vector with coordinates.
It can be proven that the gradient of a function characterizes the direction of the fastest growth of the level of a function at a point.
For example, for the function z = 2x 1 + x 2 (see Figure 5.8), the gradient at any point will have coordinates (2; 1). You can construct it on a plane in various ways, taking any point as the beginning of the vector. For example, you can connect point (0; 0) to point (2; 1), or point (1; 0) to point (3; 1), or point (0; 3) to point (2; 4), or so on. .P. (See Figure 5.8). All vectors constructed in this way will have coordinates (2 – 0; 1 – 0) = = (3 – 1; 1 – 0) = (2 – 0; 4 – 3) = (2; 1).
From Figure 5.8 it is clearly seen that the level of the function increases in the direction of the gradient, since the constructed level lines correspond to the level values 4 > 3 > 2.
Figure 5.8 - Gradient of function z= 2x 1 + x 2
Let's consider another example - the function z = 1/(x 1 x 2). The gradient of this function will no longer always be the same at different points, since its coordinates are determined by the formulas (-1/(x 1 2 x 2); -1/(x 1 x 2 2)).
Figure 5.9 shows the function level lines z = 1/(x 1 x 2) for levels 2 and 10 (the straight line 1/(x 1 x 2) = 2 is indicated by a dotted line, and the straight line 1/(x 1 x 2) = 10 is solid line).
Figure 5.9 - Gradients of the function z= 1/(x 1 x 2) at various points
Take, for example, the point (0.5; 1) and calculate the gradient at this point: (-1/(0.5 2 *1); -1/(0.5*1 2)) = (-4; - 2). Note that the point (0.5; 1) lies on the level line 1/(x 1 x 2) = 2, because z=f(0.5; 1) = 1/(0.5*1) = 2. To draw the vector (-4; -2) in Figure 5.9, connect the point (0.5; 1) with the point (-3.5; -1), because (-3.5 – 0.5; -1 - 1) = (-4; -2).
Let's take another point on the same level line, for example, point (1; 0.5) (z=f(1; 0.5) = 1/(0.5*1) = 2). Let's calculate the gradient at this point (-1/(1 2 *0.5); -1/(1*0.5 2)) = (-2; -4). To depict it in Figure 5.9, we connect the point (1; 0.5) with the point (-1; -3.5), because (-1 - 1; -3.5 - 0.5) = (-2; - 4).
Let's take another point on the same level line, but only now in a non-positive coordinate quarter. For example, point (-0.5; -1) (z=f(-0.5; -1) = 1/((-1)*(-0.5)) = 2). The gradient at this point will be equal to (-1/((-0.5) 2 *(-1)); -1/((-0.5)*(-1) 2)) = (4; 2). Let's depict it in Figure 5.9 by connecting the point (-0.5; -1) with the point (3.5; 1), because (3.5 – (-0.5); 1 – (-1)) = (4 ; 2).
It should be noted that in all three cases considered, the gradient shows the direction of growth of the function level (towards the level line 1/(x 1 x 2) = 10 > 2).
It can be proven that the gradient is always perpendicular to the level line (level surface) passing through a given point.
Extrema of a function of several variables
Let's define the concept extremum for a function of many variables.
A function of many variables f(X) has at point X (0) maximum (minimum), if there is a neighborhood of this point such that for all points X from this neighborhood the inequalities f(X)f(X (0)) () are satisfied.
If these inequalities are satisfied as strict, then the extremum is called strong, and if not, then weak.
Note that the extremum defined in this way is local character, since these inequalities are satisfied only for a certain neighborhood of the extremum point.
A necessary condition for a local extremum of a differentiable function z=f(x 1, . . ., x n) at a point is the equality to zero of all first-order partial derivatives at this point: 
.
The points at which these equalities hold are called stationary.
In another way, the necessary condition for an extremum can be formulated as follows: at the extremum point, the gradient is zero. A more general statement can also be proven: at the extremum point, the derivatives of the function in all directions vanish.
Stationary points should be subjected to additional research to determine whether sufficient conditions for the existence of a local extremum are met. To do this, determine the sign of the second order differential. If for any , not simultaneously equal to zero, it is always negative (positive), then the function has a maximum (minimum). If it can go to zero not only with zero increments, then the question of the extremum remains open. If it can take both positive and negative values, then there is no extremum at a stationary point.
In the general case, determining the sign of the differential is a rather complex problem, which we will not consider here. For a function of two variables, it can be proven that if at a stationary point 
, then the extremum is present. In this case, the sign of the second differential coincides with the sign 
, i.e. If 
, then this is the maximum, and if 
, then this is the minimum. If 
, then there is no extremum at this point, and if 
, then the question of the extremum remains open.
Example 1. Find the extrema of the function 
.
Let's find partial derivatives using the logarithmic differentiation method.
ln z = ln 2 + ln (x + y) + ln (1 + xy) – ln (1 + x 2) – ln (1 + y 2)
Likewise 
.
Let's find stationary points from the system of equations:
Thus, four stationary points have been found (1; 1), (1; -1), (-1; 1) and (-1; -1).
Let's find the second order partial derivatives:
ln (z x `) = ln 2 + ln (1 - x 2) -2ln (1 + x 2)
Likewise 
;
.
Because 
, expression sign 
depends only on 
. Note that in both of these derivatives the denominator is always positive, so you can only consider the sign of the numerator, or even the sign of the expressions x(x 2 – 3) and y(y 2 – 3). Let us define it at each critical point and check that the sufficient condition for the extremum is satisfied.
For point (1; 1) we get 1*(1 2 – 3) = -2< 0. Т.к. произведение
двух negative numbers
> 0, and 
<
0, в точке (1; 1) можно найти максимум. Он
равен
=
2*(1 + 1)*(1 +1*1)/((1 +1 2)*(1 +1 2)) =
= 8/4
= 2.
For point (1; -1) we get 1*(1 2 – 3) = -2< 0 и (-1)*((-1) 2 – 3)
= 2 >0. Because product of these numbers 
< 0, в этой точке экстремума нет.
Аналогично можно показать, что нет
экстремума в точке (-1; 1).
For the point (-1; -1) we get (-1)*((-1) 2 – 3) = 2 > 0. Because product of two positive numbers
> 0, and 
> 0, at the point (-1; -1) the minimum can be found. It is equal to 2*((-1) + (-1))*(1 +(-1)*(-1))/((1 +(-1) 2)*(1 +(-1) 2) ) = -8/4 = = -2.
Find global maximum or minimum (the largest or smallest value of a function) is somewhat more complex than local extremum, since these values can be achieved not only at stationary points, but also at the boundary of the definition domain. It is not always easy to study the behavior of a function at the boundary of this region.
GRADIENT FUNCTION u = f(x, y, z), given in some region. space (X Y Z), There is vector with projections denoted by the symbols: grad 
Where i, j, k- coordinate unit vectors. G. f. - there is a point function (x, y, z), i.e. it forms a vector field. Derivative in the direction of the G. f. at this point reaches highest value and is equal to: 
The direction of the gradient is the direction of the fastest increase in the function. G. f. at a given point is perpendicular to the level surface passing through this point. Efficiency of using G. f. during lithological studies it was shown in the study of aeolian exc. Central Karakum.
Geological Dictionary: in 2 volumes. - M.: Nedra. Edited by K. N. Paffengoltz et al.. 1978 .
See what "GRADIENT FUNCTION" is in other dictionaries:
This article is about the mathematical characteristic; about the filling method, see: Gradient (computer graphics) ... Wikipedia
- (lat.). Differences in barometric and thermometric readings in different areas. Dictionary of foreign words included in the Russian language. Chudinov A.N., 1910. GRADIENT is the difference in the readings of a barometer and thermometer at the same moment... ... Dictionary of foreign words of the Russian language
gradient- Changing the value of a certain quantity per unit distance in a given direction. Topographic gradient is the change in terrain elevation over a horizontally measured distance. Topics: relay protection EN gradient of the differential protection tripping characteristic … Technical Translator's Guide
Gradient- a vector directed in the direction of the fastest increase in the function and equal in magnitude to its derivative in this direction: where the symbols ei denote the unit vectors of the coordinate axes (orts) ... Economic and mathematical dictionary
One of the basic concepts of vector analysis and the theory of nonlinear mappings. The gradient of the scalar function of the vector argument from the Euclidean space E n is called. derivative of the function f(t). with respect to the vector argument t, that is, an n-dimensional vector with... ... Mathematical Encyclopedia
Physiological gradient- – a value reflecting a change in a function indicator depending on another value; e.g., partial pressure gradient - the difference in partial pressures that determines the diffusion of gases from the alveoli (accini) into the blood and from the blood into ... ... Glossary of terms on the physiology of farm animals
I Gradient (from Latin gradiens, gender gradientis walking) A vector showing the direction of the fastest change of some quantity, the value of which changes from one point in space to another (see Field theory). If the value... ... Great Soviet Encyclopedia
Gradient- (from the Latin gradiens walking, walking) (in mathematics) a vector showing the direction of the fastest increase of a certain function; (in physics) a measure of increase or decrease in space or on a plane of any physical quantity by unit... ... The beginnings of modern natural science
Books
- Methods for solving some problems in selected sections of higher mathematics. Workshop, Konstantin Grigorievich Klimenko, Galina Vasilievna Levitskaya, Evgeniy Alexandrovich Kozlovsky. This workshop discusses methods for solving certain types of problems from such sections of the generally accepted course. mathematical analysis, as the limit and extremum of a function, gradient and derivative...
Some concepts and terms are used within a purely narrow framework. Other definitions are found in areas that are sharply opposed. For example, the concept of “gradient” is used by a physicist, a mathematician, and a manicurist or Photoshop specialist. What is gradient as a concept? Let's figure it out.
What do dictionaries say?
Special thematic dictionaries interpret what a “gradient” is in relation to their specifics. Translated from Latin, this word means “the one who goes, grows.” And Wikipedia defines this concept as “a vector indicating the direction of increase in a quantity.” IN explanatory dictionaries we see the meaning of this word as “a change in any quantity by one value.” A concept can have both quantitative and qualitative meaning.
In short, it is a smooth gradual transition of any value by one value, a progressive and continuous change in quantity or direction. The vector is calculated by mathematicians and meteorologists. This concept is used in astronomy, medicine, art, and computer graphics. A similar term defines completely different types of activities.
Mathematical functions
What is the gradient of a function in mathematics? This indicates the direction of growth of a function in a scalar field from one value to another. The magnitude of the gradient is calculated using partial derivatives. To determine the fastest direction of growth of a function, two points are selected on the graph. They define the beginning and end of the vector. The rate at which a value grows from one point to another is the magnitude of the gradient. Mathematical functions, based on calculations of this indicator, are used in vector computer graphics, the objects of which are graphic images of mathematical objects.
What is a gradient in physics?
The concept of gradient is common in many branches of physics: gradient of optics, temperature, speed, pressure, etc. In this branch, the concept denotes a measure of increase or decrease of a value by one. It is calculated by calculations as the difference between two indicators. Let's look at some of the values in more detail.
What is a potential gradient? In work with electrostatic field two characteristics are determined: tension (power) and potential (energy). These different sizes associated with the environment. And although they determine different characteristics, still have a connection with each other.
To determine the strength of the force field, the potential gradient is used - a value that determines the rate of change in the potential in the direction power line. How to calculate? Potential difference between two points electric field is calculated from a known voltage using the voltage vector, which is equal to the potential gradient.
Terms of meteorologists and geographers
For the first time, the concept of gradient was used by meteorologists to determine changes in the magnitude and direction of various meteorological indicators: temperature, pressure, wind speed and strength. It is a measure of quantitative changes in various quantities. Maxwell introduced the term into mathematics much later. In definition weather conditions There are concepts of vertical and horizontal gradients. Let's take a closer look at them.
What is a vertical temperature gradient? This is a value that shows the change in indicators, calculated at a height of 100 m. It can be either positive or negative, in contrast to horizontal, which is always positive.
The gradient shows the magnitude or angle of slope on the ground. It is calculated as the ratio of the height to the length of the projection of the path in a certain section. Expressed as a percentage.
Medical indicators
The definition of “temperature gradient” can also be found among medical terms. It shows the difference in the corresponding indicators of internal organs and body surface. In biology, a physiological gradient records changes in the physiology of any organ or organism as a whole at any stage of its development. In medicine, the metabolic indicator is the intensity of metabolism.
Not only physicists, but also doctors use this term in their work. What is a pressure gradient in cardiology? This concept defines the difference in blood pressure in any interconnected parts of the cardiovascular system.
A decreasing gradient of automaticity is an indicator of a decrease in the frequency of excitations of the heart in the direction from its base to the top, occurring automatically. In addition, cardiologists identify the location of arterial damage and its degree by monitoring the difference in the amplitudes of systolic waves. In other words, using the amplitude gradient of the pulse.
What is a velocity gradient?
When they talk about the rate of change of a certain quantity, they mean by this the speed of change in time and space. In other words, the speed gradient determines the change in spatial coordinates in relation to time indicators. This indicator is calculated by meteorologists, astronomers, and chemists. The shear rate gradient of liquid layers is determined in the oil and gas industry to calculate the rate of rise of liquid through a pipe. This indicator of tectonic movements is the area of calculations of seismologists.
Economic functions
Economists widely use the concept of gradient to substantiate important theoretical conclusions. When solving consumer problems, a utility function is used to help represent preferences from a set of alternatives. "Budget constraint function" is a term used to refer to a set of consumption bundles. Gradients in this area are used to calculate optimal consumption.
Color gradient
The term "gradient" is familiar creative individuals. Although they are far from exact sciences. What is a gradient for a designer? Since in the exact sciences it is a gradual increase in value by one, so in color this indicator denotes a smooth, extended transition of shades of the same color from lighter to darker, or vice versa. Artists call this process “stretching.” It is also possible to switch to different accompanying colors in the same range.
Gradient stretches of shades in painting rooms have taken a strong position among design techniques. The new-fashioned ombre style - a smooth flow of shade from light to dark, from bright to pale - effectively transforms any room in the home or office.
Opticians use special lenses to sunglasses. What is a gradient in glasses? This is the making of a lens in a special way, when from top to bottom the color goes from darker to darker light shade. Products made using this technology protect the eyes from solar radiation and allow you to view objects even in very bright light.
Color in web design
Those who are involved in web design and computer graphics are familiar with universal tool“gradient”, with the help of which a wide variety of effects are created. Color transitions are transformed into highlights, a bizarre background, and three-dimensionality. Manipulating shades and creating light and shadow gives volume to vector objects. For these purposes, several types of gradients are used:
- Linear.
- Radial.
- Cone-shaped.
- Mirror.
- Diamond-shaped.
- Noise gradient.
Gradient beauty
For visitors to beauty salons, the question of what a gradient is will not come as a surprise. True, even in this case, knowledge of mathematical laws and fundamentals of physics is not necessary. We are still talking about color transitions. The objects of the gradient are hair and nails. The ombre technique, which means “tone” in French, came into fashion from sports lovers of surfing and other beach activities. Naturally bleached and regrown hair has become a hit. Fashionistas began to specially dye their hair with a barely noticeable transition of shades.
The ombre technique has not passed by nail salons. A gradient on the nails creates a color with a gradual lightening of the plate from the root to the edge. Masters offer horizontal, vertical, with a transition and other varieties.
Needlework
Needlewomen are familiar with the concept of “gradient” from one more side. A similar technique is used in creating things self made in decoupage style. In this way, new antique things are created, or old ones are restored: chests of drawers, chairs, chests, etc. Decoupage involves applying a pattern using a stencil, the basis for which is a color gradient as a background.
Fabric artists have adopted this method of dyeing for new models. Dresses with gradient colors have conquered the catwalks. The fashion was picked up by needlewomen - knitters. Knitted items with smooth transition the colors are a hit.
To summarize the definition of “gradient,” we can say about a very broad area of human activity in which this term has a place. Replacement with the synonym “vector” is not always suitable, since a vector is still a functional, spatial concept. What defines the generality of the concept is a gradual change in a certain quantity, substance, physical parameter by one over a certain period. In color it is a smooth transition of tone.