Let we have a surface given by an equation of the form

We introduce the following definition.

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

tangent to some 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, there will, in general, 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 the parametric equations

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

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

The projections of this vector depend on - the coordinates of the point Р; note that since the point P is ordinary, these projections at the point P do not vanish at the same time, 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 Р.

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

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

It follows from the last equality that the LG vector and the tangent vector to the curve (2) at the point P are perpendicular. The above reasoning is valid for any curve (2) passing through the point 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 vector LG. The theorem has been proven.

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

Note that the tangent plane may not exist at singular points of the surface. At such points, the tangent lines to the surface may not lie in the same plane. So, 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 the vector (4), then, consequently, its equation has the form

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

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

her right part is the total differential of the function . Consequently, . Thus, the total differential of a function of two variables at the 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 of 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 the vector N, then its equations will have the form

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4. THEORY OF SURFACES.

4.1 EQUATIONS OF SURFACES.

A surface in 3D space can be defined:

1) implicitly: F ( x , y , z ) =0 (4.1)

2) explicitly: z = f ( x , y ) (4.2)

3) parametrically: (4.3)

or:
(4.3’)

where are scalar arguments
sometimes called curvilinear coordinates. For example, a sphere
it is convenient to set in spherical coordinates:
.

4.2 TANGENT PLANE AND NORMAL TO THE SURFACE.

If the line lies on the surface (4.1), then the coordinates of its points satisfy the surface equation:

Differentiating this identity, we get:

(4.4)

or
(4.4 ’ )

at every point on the curve on the surface. Thus, the gradient vector at non-singular points of the surface (at which the function (4.5) is differentiable and
) is perpendicular to the tangent vectors to any lines on the surface, i.e. can be used as a normal vector to formulate the equation of the tangent plane at point M 0 (x 0 , y 0 , z 0 ) surfaces

(4.6)

and as a direction vector in the normal equation:

(4.7)

In the case of an explicit (4.2) assignment of the surface, the equations of the tangent plane and the normal, respectively, take the form:

(4.8)

and
(4.9)

In the parametric representation of the surface (4.3), the vectors
lie in the tangent plane and the equation of the tangent plane can be written as:

(4.10)

and their vector product can be taken as the directing normal vector:

and the normal equation can be written as:

(4.11)

where
- parameter values ​​corresponding to point M 0 .

In what follows, we confine ourselves to considering only those points of the surface where the vectors

are not equal to zero and are not parallel.

Example 4.1 Compose the equations of the tangent plane and the normal at the point M 0 (1,1,2) to the surface of the paraboloid of revolution
.

Solution: Since the paraboloid equation is given explicitly, according to (4.8) and (4.9) we need to find
at point M 0 :

, and at the point M 0
. Then the equation of the tangent plane at the point M 0 will take the form:

2(x -1)+2(y -1)-(z-2)=0 or 2 x +2 y -z - 2=0, and the normal equation
.

Example 4.2 Compose the equations of the tangent plane and the normal at an arbitrary point on the helicoid
, .

Solution. Here ,

Tangent plane equation:

or

Normal equations:

.

4.3 THE FIRST QUADRATIC FORM OF THE SURFACE.

If the surface is given by the equation

then the curve
on it can be given by the equation
(4.12)

Radius vector differential
along the curve corresponding to the displacement from the point M 0 to a nearby point M, is equal to

(4.13)

Because
is the differential of the curve arc corresponding to the same displacement), then

(4.14)

where .

The expression on the right side of (4.14) is called the first quadratic form of the surface and plays a huge role in the theory of surfaces.

Integrating differentialds ranging from t 0 (corresponds to point M 0 ) to t (corresponds to point M), we obtain the length of the corresponding segment of the curve

(4.15)

Knowing the first quadratic form of the surface, one can find not only the lengths, but also the angles between the curves.

If a du , dv are the differentials of curvilinear coordinates corresponding to an infinitesimal displacement along one curve, and
— on the other hand, then, taking into account (4.13):

(4.16)

Using the formula

(4.17)

the first quadratic form makes it possible to calculate the area of ​​​​a region
surfaces.

Example 4.3 On a helicoid, find the length of the helix
between two points.

Solution. Because on a helix
, then . Find at a point
the first quadratic form. Denoting andv = t , we obtain the equation of this helix in the form . Quadratic shape:

= - the first quadratic form.

Here . In formula (4.15) in this case
and arc length:

=

4.4 SECOND QUADRATIC FORM OF THE SURFACE.

Denote
- unit normal vector to the surface
:

(4.18) . (4.23)

A line on a surface is called a line of curvature if its direction at each point is the principal direction.

4.6 THE CONCEPT OF GEODETIC LINES ON THE SURFACE.

Definition 4.1 . A curve on a surface is called a geodesic if its principal normal is at every point where the curvature is nonzero, coincides with the normal to the surface.

Through each point of the surface in any direction passes, and only one geodesic. On a sphere, for example, great circles are geodesics.

A parametrization of a surface is called semi-geodesic if one family of coordinate lines consists of geodesics and the other is orthogonal to it. For example, on the sphere meridians (geodesics) and parallels.

A geodesic on a sufficiently small segment is the shortest among all curves close to it connecting the same points.

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,

(Fig. 1).

Non-zero vector

→

n

called normal vector to the surface at point A if

lim B→A j = π 2 .

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

1. 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, a point on the surface is called singular 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

→ 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)

is normal to this surface at the 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.: MEI Publishing House, 2002 (p. 128).

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

Canonical normal equations can be represented as

x − x0 F "x (x 0, y 0, z 0) = y − y0 F "y (x 0 , y 0 , z 0 ) = z−z0 F "z (x 0, y 0, z 0) .

(2)

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

From this definition it follows that tangent plane equation looks like:

(3)

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

geometric sense total differential functions 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 the 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 during the transition from point a (x 0 , y 0 ) to an arbitrary point p (x , y) is B Q (Fig. 2). The corresponding applique increment is

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

Here on the right side is the differential d z of the function z = f (x, y) at the point a (x 0 , x 0 ). Consequently,
d f (x 0 , y 0 ). is the increment of the applicate 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 = f (x 0, y 0 )).

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

Consider geometric applications of the derivative of a function of several variables. Let the function of two variables be given implicitly: . This function is represented in the domain of its definition by a certain surface (Sec. 5.1). Take an arbitrary point on the given 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 these 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.

A set of curves can be drawn through a point chosen on the surface, to each of which a tangent can be drawn.

Definition 5.8.1 . The plane in which all the tangent lines are located to the lines on the surface passing through some point is called the tangent plane to the given 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 a section of a given surface by planes, (Fig. 5.8.1).

Let's write the equation of the tangent line to the curve lying at the intersection of the surface and the plane. Since this curve lies in the coordinate system, then 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)

We use the expression for the derivative implicitly given function(clause 5.7). Then , a . 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 but the equations of lines in the 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, consequently, to the tangent plane:

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

Definition 5.8.2 . Straight line through a point surface perpendicular to the tangent plane at that 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 given in space that occupies part or all of this space. Let each point of this area, according to some law, be associated with some scalar value (number).

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

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

It should be emphasized that the value of the scalar field depends only on the position of the point in the region , but does not depend on the choice of the 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.

Non-stationary 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 altitude field above sea level.

Geometrically, scalar fields are often depicted 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 value is called a level surface or an equipotential surface. In the plane case for a scalar field, this set is called the level line or equipotential line.

Obviously, the level surface equation has the form , level lines – . Giving different values ​​to the constant in these equations, we obtain a family of surfaces or level lines. For example, (embedded spheres with different radii) or (family of ellipses).

As examples of level lines from physics, one can cite isotherms (lines of equal temperatures), isobars (lines of equal pressures); from geodesy - lines of equal heights, etc.

The graph of a function of 2 variables z = f(x,y) is a surface projected onto the XOY plane into the domain of the function D.
Consider the surface σ , given by the equation z = f(x,y) , where f(x,y) is a differentiable function, and let M 0 (x 0 ,y 0 ,z 0) be a fixed point on the surface σ , i.e. z0 = f(x0,y0). Appointment. The online calculator is designed to find tangent plane and surface normal equations. The decision is made in Word format. If you need to find the equation of the tangent to the curve (y = f(x)), then you need to use this service.

Function entry rules:

Function entry rules:

Tangent plane to surface σ at her point M 0 is the plane in which the tangents to all curves drawn on the surface lie σ through a point M 0 .
The equation of the tangent plane to the surface given by the equation z = f(x,y) at the point M 0 (x 0 ,y 0 ,z 0) has the form:

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

The vector is called the surface normal vector σ at the point M 0 . The normal vector is perpendicular to the tangent plane.
Normal to the surface σ at the point M 0 is a straight line passing through this point and having the direction of the vector N.
The canonical equations of the normal to the surface given by the equation z = f(x,y) at the point M 0 (x 0 ,y 0 ,z 0), where z 0 = f(x 0 ,y 0), have the form:

Example #1. The surface is given by the equation x 3 +5y . Find the equation of the tangent plane to the surface at the point M 0 (0;1).
Solution. Let's write the tangent equations in general form: z - z 0 \u003d f "x (x 0, y 0, z 0) (x - x 0) + f" y (x 0, y 0, z 0) (y - y 0 )
By the condition of the problem x 0 = 0, y 0 = 1, then z 0 = 5
Find the partial derivatives of the function z = x^3+5*y:
f" x (x, y) = (x 3 +5 y)" x = 3 x 2
f" x (x, y) = (x 3 +5 y)" y = 5
At the point M 0 (0.1), the values ​​of partial derivatives:
f"x(0;1) = 0
f" y (0; 1) = 5
Using the formula, we obtain the equation of the tangent plane to the surface at point M 0: z - 5 = 0(x - 0) + 5(y - 1) or -5 y + z = 0

Example #2. The surface is given implicitly y 2 -1/2*x 3 -8z. Find the equation of the tangent plane to the surface at the point M 0 (1;0;1).
Solution. We find partial derivatives of the function. Since the function is given in an implicit form, we are looking for derivatives by the formula:

For our function:

Then:

At the point M 0 (1,0,1) the values ​​of partial derivatives:
f "x (1; 0; 1) \u003d -3 / 16
f"y(1;0;1) = 0
Using the formula, we obtain the equation of the tangent plane to the surface at point M 0: z - 1 \u003d -3 / 16 (x - 1) + 0 (y - 0) or 3 / 16 x + z- 19 / 16 \u003d 0

Example. Surface σ given by the equation z= y/x + xy – 5x 3 . Find the equation of the tangent plane and normal to the surface σ at the point M 0 (x 0 ,y 0 ,z 0) belonging to it if x 0 = –1, y 0 = 2.
Let's find the partial derivatives of the function z= f(x,y) = y/x + xy – 5x 3:
f x '( x,y) = (y/x + xy – 5x 3)' x \u003d - y / x 2 + y – 15x 2 ;
f y' ( x,y) = (y/x + xy – 5x 3)' y = 1/x + x.
Dot M 0 (x 0 ,y 0 ,z 0) belongs to the surface σ , so we can calculate z 0 , substituting the given x 0 = -1 and y 0 = 2 into the surface equation:

z= y/x + xy – 5x 3

z 0 = 2/(-1) + (–1) 2 – 5 (–1) 3 = 1.
At the point M 0 (–1, 2, 1) values ​​of partial derivatives:
f x '( M 0) = –1/(-1) 2 + 2 – 15(–1) 2 = –15; fy'( M 0) = 1/(-1) – 1 = –2.
Using formula (5), we obtain the equation of the tangent plane to the surface σ at the point M 0:
z – 1= –15(x + 1) – 2(y – 2) z – 1= –15x – 15 – 2y + 4 15x + 2y + z + 10 = 0.
Using formula (6), we obtain canonical equations surface normals σ at the point M 0: .
Answers: tangent plane equation: 15 x + 2y + z+ 10 = 0; normal equations: .

Example #1. Given a function z \u003d f (x, y) and two points A (x 0, y 0) and B (x 1, y 1). Required: 1) calculate the value z 1 of the function at point B; 2) calculate the approximate value z 1 of the function at point B based on the value z 0 of the function at point A, replacing the increment of the function during the transition from point A to point B with a differential; 3) compose the equation of the tangent plane to the surface z = f(x,y) at the point C(x 0 ,y 0 ,z 0).
Solution.
We write the tangent equations in general form:
z - z 0 \u003d f "x (x 0, y 0, z 0) (x - x 0) + f" y (x 0, y 0, z 0) (y - y 0)
According to the condition of the problem x 0 = 1, y 0 = 2, then z 0 = 25
Find the partial derivatives of the function z = f(x,y)x^2+3*x*y*+y^2:
f "x (x, y) \u003d (x 2 +3 x y + y 2)" x \u003d 2 x + 3 y 3
f "x (x, y) \u003d (x 2 +3 x y + y 2)" y \u003d 9 x y 2
At the point M 0 (1.2), the values ​​of partial derivatives:
f" x (1; 2) = 26
f" y (1; 2) = 36
Using the formula, we obtain the equation of the tangent plane to the surface at the point M 0:
z - 25 = 26(x - 1) + 36(y - 2)
or
-26x-36y+z+73 = 0

Example #2. Write the equations of the tangent plane and the normal to the elliptical paraboloid z = 2x 2 + y 2 at the point (1;-1;3).