magnitude gradient. function gradient

It is known from a school mathematics course 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 also be extended to an n-dimensional space (instead of two coordinates there will be n coordinates).

Gradient gradz function z=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 proved that the gradient of a function characterizes the direction of the fastest growth of the level of the function at a point.

For example, for the function z \u003d 2x 1 + x 2 (see Figure 5.8), the gradient at any point will have coordinates (2; 1). It can be built 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 t .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).

Figure 5.8 clearly shows that the level of the function grows in the direction of the gradient, since the constructed level lines correspond to the level values ​​4 > 3 > 2.

Figure 5.8 - Gradient of the function z \u003d 2x 1 + x 2

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 level lines of the function z= 1/(x 1 x 2) for levels 2 and 10 (the line 1/(x 1 x 2) = 2 is indicated by a dotted line, and the line 1/(x 1 x 2) = 10 is solid line).

Figure 5.9 - Gradients of the function z \u003d 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)) \u003d (-4; - 2). Note that the point (0.5; 1) lies on the level line 1 / (x 1 x 2) \u003d 2, because z \u003d f (0.5; 1) \u003d 1 / (0.5 * 1) \u003d 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). 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 one more 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 (-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 level of the function (toward the level line 1/(x 1 x 2) = 10 > 2).

It can be proved that the gradient is always perpendicular to the level line (level surface) passing through the given point.

Extrema of a function of several variables

Let's define the concept extremum for a function of many variables.

The function of many variables f(X) has at the point X (0) maximum (minimum), if there is such a neighborhood of this point that for all points X from this neighborhood the inequalities f(X)f(X (0)) () hold.

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 hold only for some 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 equal to zero. It is also possible to prove a more general statement - at the extremum point, the derivatives of the function in all directions vanish.

Stationary points should be subjected to additional studies - whether sufficient conditions for the existence of a local extremum are satisfied. To do this, determine the sign of the second-order differential. If for any that are not simultaneously equal to zero, it is always negative (positive), then the function has a maximum (minimum). If it can vanish not only at 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 the stationary point.

In the general case, determining the sign of the differential is a rather complicated problem, which we will not consider here. For a function of two variables, one can prove that if at a stationary point
, then there is an extremum. 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 extrema of a function
.

Let's find partial derivatives by the method of logarithmic differentiation.

ln z = ln 2 + ln (x + y) + ln (1 + xy) – ln (1 + x 2) – ln (1 + y 2)

Similarly
.

Let's find stationary points from the system of equations:

Thus, four stationary points (1; 1), (1; -1), (-1; 1) and (-1; -1) are found.

Let's find partial derivatives of the second order:

ln (z x `) = ln 2 + ln (1 - x 2) -2ln (1 + x 2)

Similarly
;
.

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 determine it at each critical point and check the fulfillment of the sufficient extremum condition.

For point (1; 1) we get 1*(1 2 - 3) = -2< 0. Т.к. произведение двух отрицательных чисел
> 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 the 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. product of two positive numbers
> 0, and
> 0, at the point (-1; -1) you can find a minimum. It is equal to 2*((-1) + (-1))*(1 +(-1)*(-1))/((1 +(-1) 2)*(1 +(-1) 2) ) = -8/4 = = -2.

Find global the maximum or minimum (the largest or smallest value of the function) is somewhat more complicated than the local extremum, since these values ​​can be achieved not only at stationary points, but also at the boundary of the domain of definition. It is not always easy to study the behavior of a function on the boundary of this region.

Some concepts and terms are used strictly within narrow limits. Other definitions are found in areas that are sharply opposed. So, for example, the concept of "gradient" is used by a physicist, and a mathematician, and a specialist in manicure or "Photoshop". What is a gradient as a concept? Let's figure it out.

What do dictionaries say?

What is a "gradient" special thematic dictionaries interpret in relation to their specifics. Translated from Latin this word means - "the one that goes, grows." And "Wikipedia" defines this concept as "a vector indicating the direction of increasing magnitude." In explanatory dictionaries, we see the meaning of this word as "a change in any value by one value." The concept can carry 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, meteorologists. This concept is used in astronomy, medicine, art, computer graphics. Under the similar term completely different types of activities are defined.

Math functions

What is the gradient of a function in mathematics? This is which 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 the definition of partial derivatives. To find out the fastest direction of growth of the function on the graph, two points are selected. They define the start and end of the vector. The rate at which a value grows from one point to another is the magnitude of the gradient. Math functions, based on the 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 a gradient is common in many branches of physics: the gradient of optics, temperature, velocity, pressure, etc. In this industry, the concept denotes a measure of the increase or decrease in a value per unit. It is calculated as the difference between the two indicators. Let's consider some of the quantities in more detail.

What is a potential gradient? In working with an electrostatic field, two characteristics are determined: tension (power) and potential (energy). These different quantities are related to the environment. And although they define different characteristics, they 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 of the field line. How to calculate? Potential difference of two points electric field is calculated from the known voltage using the intensity vector, which is equal to the potential gradient.

Terms of meteorologists and geographers

For the first time, the concept of a gradient was used by meteorologists to determine the change in the magnitude and direction of various meteorological indicators: temperature, pressure, wind speed and strength. It is a measure of the quantitative change of various quantities. Maxwell introduced the term into mathematics much later. In the definition of weather conditions, there are concepts of vertical and horizontal gradients. Let's consider them in more detail.

What is a vertical temperature gradient? This is a value that shows the change in performance, calculated at a height of 100 m. It can be either positive or negative, in contrast to the horizontal, which is always positive.

The gradient shows the magnitude or angle of the slope on the ground. It is calculated as the ratio of the height to the length of the path projection on 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 the internal organs and the surface of the body. In biology, the physiological gradient fixes a change in the physiology of any organ or organism as a whole at any stage of its development. In medicine, a metabolic indicator is the intensity of metabolism.

Not only physicists, but also physicians use this term in their work. What is pressure gradient in cardiology? This concept defines the difference in blood pressure in any interconnected sections 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, which occur automatically. In addition, cardiologists identify the site of arterial damage and its degree by controlling the difference in the amplitudes of systolic waves. In other words, using the amplitude gradient of the pulse.

What is a velocity gradient?

When one speaks of the rate of change of a certain quantity, one means by this the rate of change in time and space. In other words, the velocity gradient determines the change in spatial coordinates in relation to temporal indicators. This indicator is calculated by meteorologists, astronomers, chemists. The shear rate gradient of fluid layers is determined in the oil and gas industry to calculate the rate at which a fluid rises through a pipe. Such an indicator of tectonic movements is the area of ​​\u200b\u200bcalculations by seismologists.

Economic functions

To substantiate important theoretical conclusions, the concept of a gradient is widely used by economists. When solving consumer problems, a utility function is used, which helps to represent preferences from a set of alternatives. "Budget constraint function" is a term used to refer to a set of consumer bundles. The gradients in this area are used to calculate the optimal consumptions.

color gradient

The term "gradient" is familiar to creative people. Although they are far from the 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, stretched 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 stretching of shades in the coloring of rooms has taken a strong position among design techniques. The newfangled ombre style - a smooth flow of shade from light to dark, from bright to pale - effectively transforms any room in the house and office.

Opticians use special lenses in their sunglasses. What is a gradient in glasses? This is lens making. in a special way when the color goes from darker to lighter from top to bottom. 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 engaged in web design and computer graphics are well aware of the universal tool "gradient", with which a lot of various effects are created. Color transitions are transformed into highlights, a fancy background, three-dimensionality. Hue manipulation, light and shadow creation adds volume to vector objects. For this purpose, several types of gradients are used:

• Linear.
• Radial.
• conical.
• Mirror.
• Rhomboid.
• noise gradient.

gradient beauty

For visitors to beauty salons, the question of what a gradient is will not come as a surprise. True, in this case, knowledge of mathematical laws and the foundations of physics is not necessary. It's all about color transitions. Hair and nails become the object of the gradient. The ombre technique, which means “tone” in French, came into fashion from sports lovers of surfing and other beach activities. Naturally burnt and regrown hair has become a hit. Women of fashion began to specially dye their hair with a barely noticeable transition of shades.

The ombre technique did not pass by nail salons. The gradient on the nails creates a coloration with a gradual lightening of the plate from the root to the edge. Masters offer horizontal, vertical, with a transition and other varieties.

Needlework

The concept of "gradient" is familiar to needlewomen from another side. A technique of this kind is used in the creation of handmade items in the decoupage style. In this way, new antique things are created, or old ones are restored: chests of drawers, chairs, chests, and so on. Decoupage involves applying a pattern using a stencil, which is based on a color gradient as a background.

Fabric artists have adopted dyeing in this way for new models. Dresses with gradient colors conquered the catwalks. Fashion was picked up by needlewomen - knitters. Knitwear with a smooth color transition is a success.

Summing up the definition of "gradient", we can say about a very extensive area of ​​human activity in which this term has a place. The replacement by the synonym "vector" is not always appropriate, since the vector is, after all, a functional, spatial concept. What determines the generality of the concept is a gradual change in a certain quantity, substance, physical parameter per unit over a certain period. In color, this is a smooth transition of tone.

She said that hands are a visiting card of a girl. And she was absolutely right. You can not be stylish, spectacular without the appropriate manicure, especially in our time. Women's beauty magazines are full of innovations and delight with their new products. What is a gradient, all fashionistas know. The latest trends dictate their own rules in colors. More and more bright colors, all kinds of interpretations are present in the manicure of ladies.

The concept of gradient manicure

We can say that this is the transition of one color to another - that's what the gradient on the nails is. The mixing technique allows you to achieve incredible coloring. With a smooth and accurate performance, a blurry separating segment of the newly formed shade is clearly visible. As if a shadow appeared (ombre in French, the second name for the gradient). It's beautiful and unusual. Sometimes it is difficult to choose the color of the varnish, combined with the chosen style of clothing. The technique of applying varnish on the nail plate in a gradient style is a good solution to the issue. It is unique in that you can play with the color palette in contrast.

The main types of gradient

Having an idea of ​​​​what a gradient is, you should dwell on its types. Their number is huge, and every day there are new ones. The main ones are:

Modern gradient design decoration

In the treasury of the master's creativity, one should add not only the ability to combine different color shades, but also apply a certain design to the nails. The gradient is a great opportunity to show imagination. You should follow the measure and be aware of the latest innovations in manicure art. The fashion design trend welcomes pastel colors. This is a win-win option that is suitable for all occasions. It will look harmonious with any style of clothing.

And also experts in practice use various means and methods for decoration. Drawing a picture on one or all nails is always relevant. The use of rhinestones, sequins will give the effect of solemnity and elegance.

Experienced craftsmen know what a gradient is in a fashionable interpretation. Thanks to this method, women are individual and unique. A modern gradient can be done not only in salons, but also at home. The desire to be beautiful knows no bounds.