At some point and has continuous private derivatives in it, at least one of which does not apply to zero, then in the vicinity of this point, the surface specified by equation (1) will right surface.

In addition to the above implicit way to task The surface can be determined obviousIf one of the variables, for example z, can be expressed in the rest:

Also exists parametric Method of assignment. In this case, the surface is determined by the system of equations:

Concept of a simple surface

More accurately, simple surface An image of a homeomorphic mapping is called (that is, a mutually unambiguous and mutually continuous display) of the inside of a single square. This definition can be given an analytical expression.

Suppose on the plane with the rectangular coordinate system U and V, the square is set, the coordinates of the internal points of which satisfy inequalities 0< u < 1, 0 < v < 1. Гомеоморфный образ квадрата в пространстве с прямоугольной системой координат х, у, z задаётся при помощи формул х = x(u, v), у = y(u, v), z = z(u, v) (параметрическое задание поверхности). При этом от функций x(u, v), y(u, v) и z(u, v) требуется, чтобы они были непрерывными и чтобы для различных точек (u, v) и (u", v") были различными соответствующие точки (x, у, z) и (x", у", z").

Example simple surface is half asphere. The whole sphere is not simple surface. This causes the need to further generalize the concept of the surface.

Subset of space, each point of which has a neighborhood, which is simple surface, called right surface .

SURFACE IN DIFFERENTIAL Geometry

Helicoid

Kattenoid

Metric does not define the unique surface shape. For example, a metric of helicide and a oteoid, parameterized accordingly, coincides, that is, between their regions there is a correspondence that maintains all the lengths (isometry). Properties that persist in isometric transformations are called internal geometry Surfaces. Internal geometry does not depend on the position of the surface in space and does not change when it is bent without stretching and compressing (for example, when the cylinder is bent into the cone).

Metric coefficients define not only the length of all curves, but in general, the results of all measurements inside the surface (angles, area, curvature, etc.). Therefore, everything that depends only on the metric refers to internal geometry.

Normal and Normal section

Normal vectors at surface points

One of the main characteristics of the surface is its normal - single vector, perpendicular tangent plane at a specified point:

.

Normal sign depends on the choice of coordinates.

The surface cross section of the plane containing the normal (at this point) forms some curve on the surface, which is called normal cross section Surfaces. The main standard for normal cross section coincides with the normal to the surface (with an accuracy of the sign).

If the curve on the surface is not a normal cross section, then its main normal forms a certain angle θ with the normal surface. Then Crivale k. The curve is related to the curvature k. n. Normal section (with the same tangent) Menias formula:

The coordinates of the ort of normal for different ways to task the surface are shown in the table:

	Coordinates of normal at the point of the surface
implicit task
explicit task
parametric task

Curvature

For different directions at a given surface point, a different curvature of a normal cross section is obtained, which is called normal curvature; She is attributed to the plus sign, if the main normal normal curve goes in the same direction as the normal to the surface, or minus, if the directions are opposite.

Generally speaking, at each point of the surface there are two perpendicular directions. e. 1 I. e. 2, in which normal curvature takes the minimum and maximum value; These directions are called main. The exception is the case when normal curvature in all directions of the same (for example, in the sphere or on the end of the ellipsoid of rotation), then all directions at the point are the main ones.

Surfaces with negative (left), zero (center) and positive (right) curvature.

Normal curvatures in the main directions are called the main curvators; Denote them κ 1 and κ 2. Value:

K. \u003d κ 1 κ 2

called gaussian curvature, full curvature or simply curvature Surfaces. The term is also found scalar curvaturewhich implies the result of the curvature tensor; At the same time, the scalar of curvature is twice as much as Gaussian curvature.

Gaussian curvature can be calculated through the metric, and therefore it is an object of internal geometry of surfaces (we note that the main curvatures to the internal geometry do not relate). By the sign of curvature, you can classify the surface points (see Figure). The curvature of the plane is zero. The curvature of the Radius R radius is everywhere. There is also a surface of constant negative curvature - pseudosphere.

Geodesic lines, geodesic curvature

Curve on the surface is called geodesic line, or simply geodesicIf at all its points the main normal normal to the curve coincides with the normal to the surface. Example: on the plane geodesic will be straight and segments of straight lines, on the sphere - large circles and their segments.

Equivalent definition: the geodesic line has a projection of its main normal on the touching plane there is a zero vector. If the curve is not a geodesic, then the specified projection is non-zero; Its length is called geodesic curvature k. g. curve on the surface. The ratio is:

,

where k. - curvature of this curve, k. n. - the curvature of its normal cross section with the same tangent.

Geodesic lines relate to internal geometry. List their main properties.

• Through this surface of the surface, one and only one geodesic passes in a given direction.
• On a sufficiently small section of the surface, two points can always be combined with geodesic, and with just one. Explanation: On the sphere, the opposite poles connect the infinite amounts of meridians, and two close points can be combined not only by the segment of a large circle, but also its addition to a complete circle, so that unambiguity is observed only in small.
• Geodesic is shortly. More strictly: on a small piece of surface, the shortest path between the predetermined points is geodesic.

Area

Another important attribute of the surface is her area which is calculated by the formula:

Namely, about what you see in the title. Essentially, this is a "spatial analogue" objectives of finding tangent and normal To the graph of the function of one variable, and therefore there should be no difficulties.

Let's start with the basic questions: what is a tangent plane and what is normal? Many are aware of these concepts at the level of intuition. The simplest model coming to mind is a ball on which a thin flat cardboard is lying. Cardboard is located as close as possible to the sphere and concerns it in a single point. In addition, at the touch point, it is fixed strictly up the needle.

In theory there is a rather witty determination of the tangent plane. Imagine free surface And the point belonging to it. Obviously, a lot of point passes through the point spatial lineswhich belong to this surface. Who has any associations? \u003d) ... Personally, I presented octopus. Suppose that each such line exists spatial tangent At point.

Definition 1.: tangent plane to the surface at the point is planecontaining tangents to all curves that belong to this surface and pass through the point.

Definition 2.: normal to the surface at the point is straight, passing through this point perpendicular to the tangent plane.

Simple and elegant. By the way, so that you do not die from boredom from the simplicity of the material, a little later, I will share with you one elegant secret that allows you to forget about the bunning of various definitions forever.

With working formulas and algorithm, solutions will get acquainted directly on a specific example. In the overwhelming majority of tasks, the equation of the tangent plane and the normal equation are also required:

Example 1.

Decision: if the surface is set by the equation (i.e. implicitly), The equation of the tangent plane to this surface at the point can be found according to the following formula:

Paying special attention to unusual private derivatives - their do not be confused from partial derivatives implicitly specified function (although the surface is defined). If you find these derivatives you need to be guided differentiation Rules of the Function of Three Variables, that is, when differentiated by any variable, two other letters are considered constants:

Without departing from the box office, find a private derivative at the point:

Similarly:

It was the most unpleasant moment of a solution in which an error if not allowed, it is constantly seemed. However, there is an effective receipt of the check, which I talked about in class Gradient derivative.

All "ingredients" are found and now it's a neat substitution with further simplifications:

– general equation The desired tangent plane.

I strongly recommend checking out this stage of the solution. First you need to make sure that the coordinates of the touch point are really satisfying the found equation:

- Faithful equality.

Now "remove" the coefficients of the general equation of the plane and check them for coincidence or proportionality with the corresponding values. In this case are proportional. How do you remember from course of analytical geometry, - this is vector Normal tangent plane and it's - vector guide Normal straight. Make up canonical equations Normal on the point and the guide vector:

In principle, the denominators can be reduced to the "Two", but there is no particular need for this

Answer:

The equations are not rebeling to designate some letters, however, again - why? Here and so extremely clear what's what.

The following two examples for an independent solution. Small "mathematical patter":

Example 2.

Find the equations of the tangent plane and normal to the surface at the point.

And the task, interesting from a technical point of view:

Example 3.

Make the equations of the tangent plane and normal to the surface at the point

At point.

There are all the chances not only to get confused, but also to face difficulties in writing. canonical equations are direct. And the equations are normal, as you probably understood, it is customary to record in this form. Although, due to the forgetfulness or ignorance of some nuances more than acceptable and parametric form.

Exemplary samples of finishing decisions at the end of the lesson.

Is there a tangent plane in any surface? In general, of course, no. Classic example is conical surface And the point - the tangents at this point directly form a conical surface, and, of course, are not lying in the same plane. In anything easy to make sure and analytically :.

Another source of problems is the fact non-existence any particular derivative at the point. However, this does not mean that at this point there is no single tangent plane.

But it was rather popularly popular than practically significant information, and we return to the urgent matters:

How to make equations of the tangent plane and normal at the point,
if the surface is specified by an explicit function?

Rewrite it in an implicit form:

And on the same principles find private derivatives:

Thus, the formula of the tangent plane is transformed into the following equation:

And, accordingly, canonical equations are normal:

How to guess, - This is already "real" private derivatives of two variables At the point that we used to identify the letter "Zet" and found 100,500 times.

Note that this article is enough to remember the very first formula from which if necessary, it is easy to remove everything else. (Clear, possessing the basic level of preparation). This approach should be used during the study of the exact sciences, i.e. From a minimum of information, it is necessary to strive to "pull out" the maximum of conclusions and consequences. "Reaching" and already existing knowledge to help! This principle is also useful in the fact that with a high probability will save in a critical situation when you know very little.

We will work out "modified" formulas for a pair of examples:

Example 4.

Make the equations of the tangent plane and normal to the surface At point.

The lining here turned out with the notation - now the letter denotes the point of the plane, but what to do is such a popular letter ....

Decision: The equation of the desired tangent plane will be according to the formula:

Calculate the value of the function at the point:

Calculate private derivatives of the 1st order At this point:

In this way:

Carefully, not in a rush:

We write canonical equations of normal at the point:

Answer:

And the final example for an independent decision:

Example 5.

Make the equations of the tangent plane and the normal to the surface at the point.

Final - because I actually explained all the technical moments and there is nothing to add. Even the functions proposed in this task, sadness and monotonous - almost guaranteed in practice you will get "polynomial", and in this sense, Example No. 2 with an exponent looks like "White Voron". By the way, it is much likely to meet the surface specified by the equation and this is another reason why the function entered the article by the "second number".

And finally, the promised secret: so how to avoid bunches of definitions? (I certainly do not mean the situation when a student is feverishly shaved in front of the exam)

The definition of any concept / phenomenon / object, first of all, gives the answer to the next question: what is it? (who / such / such / such). Consciously Answering this question, you must try to reflect significantsigns definite Identifying this or that concept / phenomenon / object. Yes, at first it turns out somewhat obliquely, inaccurately and excessively (the teacher will correct \u003d)), but over time, a completely decent scientific speech is developing.

Repeat on the most disturbed objects, for example, answer the question: who is such Cheburashka? Not so, everything is simple ;-) This is a fabulous character with big ears, eyes and brown wool "? Far and very far from the definition - there are few characters with such characteristics .... But it is already much closer to the definition: "Cheburashka is a character invented by the writer Edward Asspensky in 1966, which ... (transfer of the main distinctive features)". Pay attention to how competent started

Let we have a surface specified by the type equation

We introduce the following definition.

Definition 1. The straight line is called tangent to the surface at some point if it is

tangent to any curve lying on the surface and passing through the point.

Since through the point P passes the infinite number of different curves lying on the surface, and tangents to the surface passing through this point will, generally speaking, the infinite set.

We introduce the concept of special and ordinary surface points

If at the point all three derivatives are zero or at least one of these derivatives does not exist, then the point M is called a special point of the surface. If at the point all three derivatives exist and are continuous, and at least one of them is different from zero, then the point M is called an ordinary surface point.

Now we can formulate the following theorem.

Theorem. All tangent lines to this surface (1) in its ordinary point p lie in the same plane.

Evidence. Consider a certain line L (Fig. 206), passing through this point of the surface. Let the curve under consideration set by parametric equations

The tangent to the curve will be tangent to the surface. Equations of this tangent

If expressions (2) substitute in equation (1), this equation will turn into identity relative to T, since the curve (2) lies on the surface (1). Differentiating it by receiving

The projections of this vector depend on the coordinates of the point P; Note that since the point r is ordinary, then these projections at the point p simultaneously do not turn to zero and therefore

tangent to the curve passing through the point P and lying on the surface. The projections of this vector are calculated on the basis of equations (2) with the value of the parameter T corresponding to the point R.

We calculate the scalar product of vectors n and which is equal to the amount of the works of the same names:

Based on equality (3), the expression that stands in the right part is zero, therefore,

From the last equality it follows that the LG vector and the tangent vector to the curve (2) at the point p perpendicular. The argument is valid for any curve (2) passing through the P and lying on the surface. Consequently, each tangent to the surface at the point P is perpendicular to the same vector n and therefore all these tangents lie in the same plane perpendicular to the LG vector. Theorem is proved.

Definition 2. The plane in which all tangent direct lines on the surfaces passing through this point p is called a tangent plane to the surface at the point P (Fig. 207).

Note that in different points of the surface there may not be a tangent plane. At such points, tangent direct to the surface may not lie in the same plane. For example, the vertex of the conical surface is a special point.

Tangents to the conical surface at this point are not lying in the same plane (they themselves form a conical surface).

Write the equation of the tangent plane to the surface (1) in an ordinary point. Since this plane is perpendicular to the vector (4), then, therefore, its equation has the form

If the equation of the surface is specified in the form or equation of the tangent plane in this case will take the form

Comment. If we put in formula (6), then this formula will take the form

its right part is a complete differential function. Hence, . Thus, the full differential function of the two variables at the point corresponding to the increments of independent variables x and y is equal to the corresponding increment of the applicature of the tangent plane to the surface, which is a graph of this function.

About the title 3. Direct, conducted through the point of the surface (1) perpendicular to the tangent plane, is called normal to the surface (Fig. 207).

We will write the normal equations. Since its direction coincides with the direction of the vector N, then its equations will be

Equation of the normal plane

1.

4.

Tangent plane and normal surface

Let some surface be given, A is a fixed surface point and B - variable surface point,

(Fig. 1).

Nonzero vector

→

n.

called normal vector to the surface at point A, if

Lim. B → A. j \u003d. π 2 .

The surface point f (x, y, z) \u003d 0 is called ordinary, if at this point

1. private derivatives f "x, f" y, f "z are continuous;
2. (F "x) 2 + (f" y) 2 + (f "z) 2 ≠ 0.

In case of violation of at least one of these conditions, the surface point is called special point of surface .

Theorem 1.If M (x 0, y 0, z 0) - ordinary surface point F (x, y, z) \u003d 0, then vector

→ n. \u003d grad f (x 0, y 0, z 0) \u003d f "x (x 0, y 0, z 0) → i. + F "y (x 0, y 0, z 0) → j. + F "z (x 0, y 0, z 0) → k.

(1)

it is normal to this surface at point M (x 0, y 0, z 0).

Evidenceled in the book I.M. Petrushko, L.A. Kuznetsova, V.I. Prokhorenko, V.F. Safonova `` course of higher mathematics: integral calculus. Functions of several variables. Differential equations. M.: Publishing House MEI, 2002 (p. 128).

Normal to the surface In some of its point, the direct, the guide vector of which is normal to the surface at this point and which passes through this point.

Canonical equations Normal can be represented as

X - X 0 F "x (x 0, y 0, z 0) = Y - Y 0 F "y (x 0, y 0, z 0) = z - z 0 F "z (x 0, y 0, z 0) .

(2)

Tangent plane To the surface at some point is the plane, which passes through this point perpendicular to the normal surface to the surface at this point.

From this definition it follows that equation of the tangent plane It has the form:

(3)

If the surface point is particular, then at this point normal to the surface vector may not exist, and, therefore, the surface may not have the normal and tangent plane.

The geometrical meaning of the full differential function of two variables

Let the function z \u003d f (x, y) differentiate at the point A (x 0, y 0). Its schedule is the surface

f (x, y) - z \u003d 0.

Put z 0 \u003d f (x 0, y 0). Then point A (x 0, y 0, z 0) belongs to the surface.

Private derivatives F (X, Y, Z) \u003d F (X, Y) - Z essence

F "x \u003d f" x, f "y \u003d f" y, f "z \u003d - 1

and at point A (x 0, y 0, z 0)

1. they are continuous;
2. F "2 x + f" 2 y + f "2 z \u003d f" 2 x + f "2 y + 1 ≠ 0.

Consequently, A is an ordinary surface point F (x, y, z) and at this point there is a tangent plane to the surface. According to (3), the equation of the tangent plane is:

f "x (x 0, y 0) (x - x 0) + f" y (x 0, y 0) (y - y 0) - (z - z 0) \u003d 0.

A vertical displacement of the point on the tangent plane when switching from point A (x 0, y 0) into an arbitrary point P (x, y) is b Q (Fig. 2). Applied increment applicatis

(z - z 0) \u003d f "x (x 0, y 0) (x - x 0) + f" y (x 0, y 0) (y - y 0)

Here in the right part there is a differential d. z functions z \u003d f (x, y) at point A (x 0, x 0). Hence,
d. f (x 0, y 0). There is an increment of the applications of the point of the plane tangent to the graph of the function f (x, y) at the point (x 0, y 0, z 0 \u003d f (x 0, y 0)).

From the determination of the differential it follows that the distance between the point P on the function graph and the point q on the tangent plane there is an infinitely small order than the distance from point P to point a.

1 °

1 °. Equations of the tangent plane and normal for the case of an explicit surface task.

Consider one of the geometric applications of the private derivatives of the functions of two variables. Let the function z. = f (x;y) Differential in point (x 0; y 0) Some region D.Î R 2.. Dissect the surface S,picture function z, Planes x \u003d x 0 and y \u003d y 0 (Fig. 11).

Plane h. = x 0 Crossing the surface S. For some line z 0 (y) The equation of which is obtained by substitution to the expression of the original function z \u003d.=f (x;y) instead h. numbers x 0. Point M 0 (x 0;y 0f (x 0;y 0))belong Krivoy z 0 (y). By virtue of the differential function z. At point M 0 function z 0 (y) is also differentiated at the point y \u003d y 0. Consequently, at this point in the plane x \u003d x 0 To curve z 0 (y) maybe tangent l 1.

Conducting similar arguments for cross section w. = 0, We build tangent l 2. To curve z 0 (x) At point h. = x 0 - Straight 1 1 and 1 2 determine the plane called tangent plane To surface S. At point M 0.

Make its equation. As the plane passes through the point Mo (x 0;y 0;z 0), then its equation can be recorded in the form

A (x - ho) + in (y - uh) + c (z - zo) \u003d 0,

which can be rewritten:

z -z 0 \u003d a 1 (x - x 0) + b 1 (y - y 0) (1)

(separating the equation to -s and indicating ).

Find A 1. and b 1.

Equations tangents 1 1 and 1 2 have kind

respectively.

Tangent l 1. Lies in the plane A , Consequently, the coordinates of all points l 1. Satisfy equation (1). This fact can be written as a system.

Allowing this system relative to B 1, we obtain that. Conductive similar reasoning for tangential l 3.Easy to install that.

Substituting meanings A 1. and B 1 to equation (1), we obtain the desired equation of the tangent plane:

Straight, passing through the point M 0 and perpendicular to the tangent plane built at this surface of the surface is called it normal.

Using the condition perpendicularity of the direct and plane, it is easy to obtain canonical equations of normal:

Comment. The formulas of the tangent plane and normal to the surface are obtained for ordinary, i.e. not special, surface points. Point M 0 Surfaces are called special If at this point all private derivatives are zero or at least one of them does not exist. We do not consider such points.

Example. Write equations of the tangent plane and normal to the surface at its point M (2; -1; 1).

Decision. We will find private derivatives of this function and their values \u200b\u200bat the point M

Hence, applying formulas (2) and (3), we will have: z-1 \u003d 2 (x - 2) +2 (y + 1) or 2x + 2ow-Z - 1 \u003d 0 - equation of the tangent plane and - equations of normal.

2 °. Equations of the tangent plane and normal for the case of an implicit surface task.

If surface S. Posted by equation F (x; y;z) \u003d 0, then equations (2) and (3), taking into account the fact that private derivatives can be found as derivatives of an implicit function.