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

Let's consider one of the geometric applications of partial derivatives of a function of two variables. Let the function z = f (x ;y) differentiable at the point (x 0; y 0) some area DÎ R 2. Let's cut the surface S, representing the function z, planes x = x 0 And y = y 0(Fig. 11).

Plane X = x 0 intersects the surface S along some line z 0 (y ), the equation of which is obtained by substituting into the expression of the original function z ==f (x ;y) instead of X numbers x 0 . Dot M 0 (x 0 ;y 0,f (x 0 ;y 0)) belongs to the curve z 0 (y). Due to the differentiable function z at the point M 0 function z 0 (y) is also differentiable at the point y =y 0 . Therefore, at this point in the plane x = x 0 to the curve z 0 (y) a tangent can be drawn l 1.

Carrying out similar reasoning for the section at = y 0, let's build a tangent l 2 to the curve z 0 (x) at the point X = x 0 - Direct 1 1 And 1 2 define a plane called tangent plane to the surface S at the point M 0.

Let's create its equation. Since the plane passes through the point Mo(x 0 ;y 0 ;z 0), then its equation can be written as

A(x - xo) + B(y - yo) + C (z - zo) = 0,

which can be rewritten like this:

z -z 0 = A 1 (x – x 0) + B 1 (y – y 0) (1)

(dividing the equation by -C and denoting ).

We'll find A 1 and B 1.

Tangent equations 1 1 And 1 2 look like

respectively.

Tangent l 1 lies in plane a , therefore, the coordinates of all points l 1 satisfy equation (1). This fact can be written in the form of a system

Resolving this system with respect to B 1, we obtain that. Carrying out similar reasoning for the tangent l 3, it is easy to establish that .

Substituting the values A 1 and B 1 into equation (1), we obtain the required tangent plane equation:

Line passing through a point M 0 and perpendicular to the tangent plane constructed at this point on the surface is called its normal.

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

Comment. Formulas for the tangent plane and normal to the surface are obtained for ordinary, i.e., non-special, points of the surface. Dot M 0 surface is called special, if at this point all partial derivatives are equal to zero or at least one of them does not exist. We do not consider such points.

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

Solution. Let's find the partial derivatives of this function and their values ​​at point M

From here, applying formulas (2) and (3), we will have: z-1=2(x-2)+2(y+1) or 2х+2у-z-1=0- tangent plane equation and - normal equations.

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

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

Definition 1 : The tangent plane to the surface at a given point P (x 0, y 0, z 0) is a plane passing through point P and containing all the tangents constructed at point P to all possible curves on this surface passing through point P.

Let the surface s be given by the equation F (X, at, z) = 0 and point P (x 0 , y 0 , z 0) belongs to this surface. Let us select some curve on the surface L, passing through the point R.

Let X = X(t), at = at(t), z = z(t) - parametric equations of the line L.

Let's assume that: 1) function F(X, at, z) is differentiable at the point R and not all of its partial derivatives at this point are equal to zero; 2) functions X(t), at(t), z(t) are also differentiable.

Since the curve belongs to the surface s, the coordinates of any point on this curve, being substituted into the equation of the surface, will turn it into an identity. Thus, the identical equality is true: F [x(t), at(t), z (t)]= 0.

Differentiating this identity with respect to the variable t, using the chain rule, we obtain a new identical equality, valid at all points of the curve, including at the point P (x 0 , y 0 , z 0):

Let point P correspond to the parameter value t 0, that is x 0 = x (t 0), y 0 = y (t 0), z 0 = z (t 0). Then the last relation calculated at the point R, will take the form

This formula is the scalar product of two vectors. The first one is a constant vector

independent of the choice of curve on the surface.

The second vector is tangent at the point R to the line L, which means it depends on the choice of line on the surface, that is, it is a variable vector.

With the introduced notation, the equality is:

let's rewrite how.

Its meaning is this: the scalar product is equal to zero, therefore, the vectors are perpendicular. Selecting all possible curves passing through a point R on surface s, we will have different tangent vectors constructed at the point R to these lines; the vector does not depend on this choice and will be perpendicular to any of them, that is, all tangent vectors are located in the same plane, which, by definition, is tangent to the surface s, and the point R in this case it is called the tangent point. The vector is the surface normal direction vector.

Definition 2: The normal to the surface s at point P is a straight line passing through point P and perpendicular to the tangent plane constructed at this point.

We have proven the existence of a tangent plane, and, consequently, a normal to the surface. Let's write down their equations:

Equation of the tangent plane constructed at point P (x0, y0, z0) to the surface s given by the equation F(x, y, z) = 0;

Equation of the normal constructed at a point R to the surface s.

Example: Find the equation of the surface formed by the rotation of the parabola:

z 2 = 2p (y +2)

around the y axis, calculate provided that the point M(3, 1, - 3) belongs to the surface. Find the equations of the normal and tangent plane to the surface at point M.

Solution. Using the rule for writing a surface of rotation, we obtain:

z 2 + x 2 = 2p (y +2) .

Substituting the coordinates of point M into this equation, we calculate the value of the parameter p: 9 + 9 = 2р(1 + 2) . We record the final view of the surface of revolution passing through the point M:

z 2 + x 2 = 6(y +2).

Now we will find the equations of the normal and tangent plane using the formulas, for which we first calculate the partial derivatives of the function:

F(x, y) = z 2 + x 2- 6 (y +2):

Then the equation of the tangent plane takes the form 6(x - 3) - 6(y - 1) - 6(z + 3) = 0 or x - y - z - 5 = 0;

Normal plane equation

1.

4.

Tangent plane and surface normal

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

Nonzero vector

A surface point F (x, y, z) = 0 is called ordinary if at this point

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

If at least one of these conditions is violated, the surface point is called special point of the surface .

Theorem 1. If M(x 0 , y 0 , z 0 ) is an ordinary point of the surface F (x , y , z) = 0 , then the vector

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

Proof given in the book by 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 MPEI, 2002 (p. 128).

Normal to the surface at some point there is a straight line whose direction vector is normal to the surface at this point and which passes through this point.

Canonical normal equations can be represented in the form

Tangent plane to the surface at a certain point is a plane that passes through this point perpendicular to the normal to the surface at this point.

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

If a point on a surface is singular, then at that point the vector normal to the surface may not exist, and, therefore, the surface may not have a normal and a tangent plane.

Geometric meaning of the total differential of a function of two variables

Let the function z = f (x, y) be differentiable at the point a (x 0, y 0). Its graph is the surface

f (x, y) − z = 0.

Let's put z 0 = f (x 0 , y 0 ) . Then point A (x 0 , y 0 , z 0 ) belongs to the surface.

The partial derivatives of the function F (x, y, z) = f (x, y) − z are

F " x = f " x , F " y = f " y , F " z = − 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 = f "2 x + f "2 y + 1 ≠ 0.

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

f " x (x 0 , y 0 ) (x − x 0 ) + f " y (x 0 , y 0 ) (y − y 0 ) − (z − z 0 ) = 0.

The vertical displacement of a point on the tangent plane when moving from point a (x 0 , y 0 ) to an arbitrary point p (x , y) is B Q (Fig. 2). The corresponding increment of applicates is

(z − z 0 ) = f " x (x 0 , y 0 ) (x − x 0 ) + f " y (x 0 , y 0 ) (y − y 0 )

Here on the right side there is a differential d z function z = f (x, y) at point a (x 0, x 0). Hence,
d f (x 0 , y 0 ). is the increment of the applicate of a tangent plane point to the graph of the function f (x, y) at the point (x 0, y 0, z 0 = f (x 0, y 0)).

From the definition of a differential it follows that the distance between point P on the graph of a function and point Q on the tangent plane is an infinitesimal of a higher order than the distance from point p to point a.

Let us have a surface defined by an equation of the form

Let us introduce the following definition.

Definition 1. A 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 an infinite number of different curves lying on the surface pass through the point P, then, generally speaking, there will be an infinite number of tangents to the surface passing through this point.

Let us introduce the concept of singular and ordinary points of a surface

If at a point all three derivatives are equal to zero or at least one of these derivatives does not exist, then the point M is called a singular point of the surface. If at a 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 point of the surface.

Now we can formulate the following theorem.

Theorem. All tangent lines to a given surface (1) at its ordinary point P lie in the same plane.

Proof. Let us consider a certain line L on the surface (Fig. 206) passing through a given point P of the surface. Let the curve under consideration be given by parametric equations

The tangent to the curve will be the tangent to the surface. The equations of this tangent have the form

If expressions (2) are substituted into equation (1), then this equation will turn into an identity with respect to t, since curve (2) lies on surface (1). Differentiating it by we get

The projections of this vector depend on - the coordinates of point P; note that since point P is ordinary, these projections at point P do not simultaneously vanish and therefore

tangent to a curve passing through point P and lying on the surface. The projections of this vector are calculated based on equations (2) at the value of the parameter t corresponding to point P.

Let us calculate the scalar product of vectors N and which is equal to the sum of the products of projections of the same name:

Based on equality (3), the expression on the right side is equal to zero, therefore,

From the last equality it follows that the vector LG and the tangent vector to curve (2) at point P are perpendicular. The above reasoning is valid for any curve (2) passing through point P and lying on the surface. Consequently, each tangent to the surface at point P is perpendicular to the same vector N and therefore all these tangents lie in the same plane perpendicular to the vector LG. The theorem is proven.

Definition 2. The plane in which all tangent lines to the lines on the surface passing through its given point P are located is called the tangent plane to the surface at point P (Fig. 207).

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

The tangents to the conical surface at this point do not lie in the same plane (they themselves form a conical surface).

Let us write the equation of the tangent plane to the surface (1) at an ordinary point. Since this plane is perpendicular to vector (4), therefore, its equation has the form

If the equation of the surface is given in the form or the equation of the tangent plane in this case takes the form

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

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

Definition 3. A straight line drawn through a point on the surface (1) perpendicular to the tangent plane is called the normal to the surface (Fig. 207).

Let's write the normal equations. Since its direction coincides with the direction of vector N, its equations will have the form

Let's consider geometric applications of the derivative of a function of several variables. Let a function of two variables be specified implicitly: . This function in its domain of definition is represented by a certain surface (Section 5.1). Let us take an arbitrary point on this surface , in which all three partial derivatives , , exist and are continuous, and at least one of them is not equal to zero.

A point with such characteristics is called ordinary surface point. If at least one of the above requirements is not met, then the point is called special surface point.

Through a point selected on the surface, many curves can be drawn, each of which can have a tangent.

Definition 5.8.1 . The plane in which all tangent lines to the lines on the surface passing through a certain point are located is called the tangent plane to this surface at the point .

To draw a given plane, it is enough to have two tangent lines, that is, two curves on the surface. These can be curves obtained as a result of cutting a given surface with planes , (Fig. 5.8.1).

Let us write the equation of a tangent line to a curve lying at the intersection of the surface and the plane. Since this curve lies in the coordinate system, the equation of the tangent to it at the point, in accordance with paragraph 2.7, has the form:

. (5.8.1)

Accordingly, the equation of the tangent to the curve lying at the intersection of the surface and the plane in the coordinate system at the same point has the form:

. (5.8.2)

Let's use the expression for the derivative of an implicitly specified function (Section 5.7). Then, eh. Substituting these derivatives into (5.8.1) and (5.8.2), we obtain, respectively:

; (5.8.3)

. (5.8.4)

Since the resulting expressions are nothing more than equations of lines in canonical form (section 15), then from (5.8.3) we obtain the direction vector , and from (5.8.4) – . The cross product will give a vector normal to the given tangent lines, and therefore to the tangent plane:

It follows that the equation of the tangent plane to the surface at the point has the form (item 14):

Definition 5.8.2 . A straight line drawn through a point surface perpendicular to the tangent plane at this point is called the normal to the surface.

Since the direction vector of the normal to the surface coincides with the normal to the tangent plane, the normal equation has the form:

.

Scalar field

Let a region be specified in space, occupying part or all of this space. Let each point of this area, according to some law, be associated with a certain scalar quantity (number).

Definition 5.9.1 . An area in space, each point of which is associated, according to a well-known law, with a certain scalar quantity, is called a scalar field.

If some kind of coordinate system is associated with the area, for example, a rectangular Cartesian system, then each point acquires its own coordinates. In this case, the scalar quantity becomes a function of coordinates: on the plane – , in three-dimensional space – . The function itself that describes this field is often called a scalar field. Depending on the dimension of space, a scalar field can be flat, three-dimensional, etc.

It must be emphasized that the magnitude of the scalar field depends only on the position of the point in the region, but does not depend on the choice of coordinate system.

Definition 5.9.2 . A scalar field that depends only on the position of a point in the region, but does not depend on time, is called stationary.

Nonstationary scalar fields, that is, time-dependent, will not be considered in this section.

Examples of scalar fields include the temperature field, the pressure field in the atmosphere, and the height field above ocean level.

Geometrically, scalar fields are often represented using so-called lines or level surfaces.

Definition 5.9.3 . The set of all points in space at which the scalar field has the same meaning is called a level surface or equipotential surface. In the flat case for a scalar field, this set is called a level line or equipotential line.

Obviously, the level surface equation has the form , level lines – . By giving the constant different values ​​in these equations, we obtain a family of surfaces or level lines. For example, (spheres nested inside each other with different radii) or (family of ellipses).

Examples of level lines from physics include isotherms (lines of equal temperatures), isobars (lines of equal pressure); from geodesy - lines of equal heights, etc.