Tangent plane and normal to the surface.
tangent plane
Let N and N 0 be points of this surface. Let's draw a straight line NN 0. The plane that passes through the point N 0 is called tangent plane to the surface if the angle between the secant NN 0 and this plane tends to zero, when the distance NN 0 tends to zero.
Definition. Normal to the surface at point N 0 is a straight line passing through point N 0 perpendicular to the tangent plane to this surface.
At any point the surface has either only one tangent plane or does not have it at all.
If the surface is given by the equation z = f(x, y), where f(x, y) is a function differentiable at the point M 0 (x 0, y 0), the tangent plane at the point N 0 (x 0,y 0, ( x 0 ,y 0)) exists and has the equation:
The equation of the normal to the surface at this point is:
Geometric sense full differential function of two variables f(x, y) at the point (x 0, y 0) is the increment of the applicate (z coordinates) of the tangent plane to the surface when moving from the point (x 0, y 0) to the point (x 0 +x, y 0 +у).
As you can see, the geometric meaning of the total differential of a function of two variables is a spatial analogue of the geometric meaning of the differential of a function of one variable.
Example. Find the equations of the tangent plane and normal to the surface
at point M(1, 1, 1).
Tangent plane equation:
Normal equation:
20.4. Approximate calculations using total differentials.
Let the function f(x, y) be differentiable at the point (x, y). Let's find the total increment of this function:
If we substitute the expression into this formula
then we get an approximate formula:
Example. Calculate approximately the value based on the value of the function at x = 1, y = 2, z = 1.
From the given expression we determine x = 1.04 – 1 = 0.04, y = 1.99 – 2 = -0.01,
z = 1.02 – 1 = 0.02.
Let's find the value of the function u(x, y, z) =
Finding partial derivatives:
The total differential of the function u is equal to:
The exact value of this expression is 1.049275225687319176.
20.5. Partial derivatives of higher orders.
If a function f(x, y) is defined in some domain D, then its partial derivatives will also be defined in the same domain or part of it.
We will call these derivatives first order partial derivatives.
The derivatives of these functions will be second order partial derivatives.
Continuing to differentiate the resulting equalities, we obtain partial derivatives of higher orders.
Definition.
Partial derivatives of the form 
etc. are called mixed derivatives.
Theorem. If the function f(x, y) and its partial derivatives are defined and continuous at the point M(x, y) and its vicinity, then the following relation is true:
Those. partial derivatives of higher orders do not depend on the order of differentiation.
Higher order differentials are defined similarly.
…………………
Here n is the symbolic power of the derivative, which is replaced by the real power after raising the expression in brackets to it.
The geometric meaning of the total differential of a function of two variables f(x, y) at the point (x 0, y 0) is the increment of the applicate (z coordinates) of the tangent plane to the surface when moving from the point (x 0, y 0) to the point (x 0 + Dх, у 0 +Dу).
Partial derivatives of higher orders. : If a function f(x, y) is defined in some domain D, then its partial derivatives will also be defined in the same domain or part of it. We will call these derivatives partial derivatives of the first order.
The derivatives of these functions will be second order partial derivatives.
Continuing to differentiate the resulting equalities, we obtain partial derivatives of higher orders. Definition. Partial derivatives of the form 
etc. are called mixed derivatives. Schwartz's theorem:
If partial derivatives of higher orders f.m.p. are continuous, then mixed derivatives of the same order differ only in the order of differentiation = from each other.
Here n is the symbolic power of the derivative, which is replaced by the real power after raising the expression in brackets to it.
14. Equation of tangent plane and surface normal!
Let N and N 0 be points of this surface. Let's draw a straight line NN 0. The plane that passes through the point N 0 is called tangent plane to the surface if the angle between the secant NN 0 and this plane tends to zero, when the distance NN 0 tends to zero.
Definition. Normal to the surface at point N 0 is a straight line passing through point N 0 perpendicular to the tangent plane to this surface.
At any point the surface has either only one tangent plane or does not have it at all.
If the surface is given by the equation z = f(x, y), where f(x, y) is a function differentiable at the point M 0 (x 0, y 0), tangent plane at point N 0 (x 0 ,y 0, (x 0 ,y 0)) exists and has the equation:
Equation of the normal to the surface at this point:
Geometric sense the total differential of a function of two variables f(x, y) at the point (x 0, y 0) is the increment of the applicate (z coordinates) of the tangent plane to the surface when moving from the point (x 0, y 0) to the point (x 0 + Dx, y 0 +Dу).
As you can see, the geometric meaning of the total differential of a function of two variables is a spatial analogue of the geometric meaning of the differential of a function of one variable.
16. Scalar field and its characteristics. Lines of the level, derivatives in direction, gradient of the scalar field.
If each point in space is associated with a scalar quantity, then a scalar field arises (for example, a temperature field, a field electric potential). If Cartesian coordinates are entered, then they also denote or 
The field can be flat if it is central 
(spherical) if 
cylindrical if 
Level surfaces and lines: The properties of scalar fields can be visually studied using level surfaces. These are surfaces in space on which it takes on a constant value. Their equation is: 
. In a flat scalar field, level lines are curves on which the field takes a constant value: 
In some cases, level lines can degenerate into points, and level surfaces into points and curves.
Directional derivative and gradient of a scalar field:
Let the unit vector with coordinates be a scalar field. The directional derivative characterizes the change in the field in a given direction and is calculated using the formula The directional derivative is scalar product vector and vector with coordinates 
, which is called the gradient of the function and is denoted . Since 
, where the angle between and , then the vector indicates the direction of the fastest increase in the field and its modulus is equal to the derivative in this direction. Since the components of the gradient are partial derivatives, it is not difficult to obtain the following properties of the gradient:
17. Extrema of f.m.p. Local extremum of f.m.p., necessary and sufficient conditions for its existence. The largest and smallest value of f.m.p. in limited closed area.
Let the function z = ƒ(x;y) be defined in some domain D, point N(x0;y0)
A point (x0;y0) is called a maximum point of the function z=ƒ(x;y) if there is a d-neighborhood of the point (x0;y0) such that for each point (x;y) different from (xo;yo), from this neighborhood the inequality ƒ(x;y) holds<ƒ(хо;уо). Аналогично определяется точка минимума функции: для всех точек (х; у), отличных от (х0;у0), из d-окрестности точки (хо;уо) выполняется неравенство: ƒ(х;у)>ƒ(x0;y0). The value of the function at the point of maximum (minimum) is called the maximum (minimum) of the function. The maximum and minimum of a function are called its extrema. Note that, by definition, the extremum point of the function lies inside the domain of definition of the function; maximum and minimum have a local (local) character: the value of the function at the point (x0; y0) is compared with its values at points sufficiently close to (x0; y0). In region D, a function may have several extrema or none.
Necessary(1) and sufficient(2) conditions for existence:
(1) If at the point N(x0;y0) the differentiable function z=ƒ(x;y) has an extremum, then its partial derivatives at this point are equal to zero: ƒ"x(x0;y0)=0, ƒ"y(x0;y0 )=0. Comment. A function can have an extremum at points where at least one of the partial derivatives does not exist. The point at which the first order partial derivatives of the function z ≈ ƒ(x; y) are equal to zero, i.e. f"x=0, f"y=0, is called a stationary point of the function z.
Stationary points and points at which at least one partial derivative does not exist are called critical points
(2)
Let the function ƒ(x;y) at a stationary point (xo; y) and some of its neighborhood have continuous partial derivatives up to the second order inclusive. Let us calculate at the point (x0;y0) the values A=f""xx(x0;y0), B=ƒ""xy(x0;y0), C=ƒ""yy(x0;y0). Let's denote 
Then:
1. if Δ > 0, then the function ƒ(x;y) at the point (x0;y0) has an extremum: maximum if A< 0; минимум, если А > 0;
2. if Δ< 0, то функция ƒ(х;у) в точке (х0;у0) экстремума не имеет.
3. In the case of Δ = 0, there may or may not be an extremum at the point (x0;y0). More research is needed.
$E \subset \mathbb(R)^(n)$. They say $f$ has local maximum at the point $x_(0) \in E$, if there is a neighborhood $U$ of the point $x_(0)$ such that for all $x \in U$ the inequality $f\left(x\right) \leqslant f is satisfied \left(x_(0)\right)$.
The local maximum is called strict , if the neighborhood $U$ can be chosen so that for all $x \in U$ different from $x_(0)$ there is $f\left(x\right)< f\left(x_{0}\right)$.
Definition
Let $f$ be real function on the open set $E \subset \mathbb(R)^(n)$. They say $f$ has local minimum at the point $x_(0) \in E$, if there is a neighborhood $U$ of the point $x_(0)$ such that for all $x \in U$ the inequality $f\left(x\right) \geqslant f is satisfied \left(x_(0)\right)$.
A local minimum is called strict if a neighborhood $U$ can be chosen so that for all $x \in U$ different from $x_(0)$ there is $f\left(x\right) > f\left(x_( 0)\right)$.
Local extremum combines the concepts of local minimum and local maximum.
Theorem ( necessary condition extremum of the differentiable function)
Let $f$ be a real function on the open set $E \subset \mathbb(R)^(n)$. If at the point $x_(0) \in E$ the function $f$ has a local extremum at this point, then $$\text(d)f\left(x_(0)\right)=0.$$ Equal to zero differential is equivalent to the fact that all are equal to zero, i.e. $$\displaystyle\frac(\partial f)(\partial x_(i))\left(x_(0)\right)=0.$$
In the one-dimensional case this is – . Let us denote $\phi \left(t\right) = f \left(x_(0)+th\right)$, where $h$ is an arbitrary vector. The function $\phi$ is defined for values of $t$ that are sufficiently small in absolute value. In addition, it is differentiable with respect to , and $(\phi)’ \left(t\right) = \text(d)f \left(x_(0)+th\right)h$.
Let $f$ have a local maximum at point x $0$. This means that the function $\phi$ at $t = 0$ has a local maximum and, by Fermat’s theorem, $(\phi)’ \left(0\right)=0$.
So, we got that $df \left(x_(0)\right) = 0$, i.e. function $f$ at point $x_(0)$ is equal to zero on any vector $h$.
Definition
Points at which the differential is zero, i.e. those in which all partial derivatives are equal to zero are called stationary. Critical points functions $f$ are those points at which $f$ is not differentiable or is equal to zero. If the point is stationary, then it does not follow from this that the function has an extremum at this point.
Example 1.
Let $f \left(x,y\right)=x^(3)+y^(3)$. Then $\displaystyle\frac(\partial f)(\partial x) = 3 \cdot x^(2)$,$\displaystyle\frac(\partial f)(\partial y) = 3 \cdot y^(2 )$, so $\left(0,0\right)$ is a stationary point, but the function has no extremum at this point. Indeed, $f \left(0,0\right) = 0$, but it is easy to see that in any neighborhood of the point $\left(0,0\right)$ the function takes both positive and negative values.
Example 2.
The function $f \left(x,y\right) = x^(2) − y^(2)$ has a stationary point at its origin, but it is clear that there is no extremum at this point.
Theorem ( sufficient condition extremum).
Let the function $f$ be twice continuously differentiable on the open set $E \subset \mathbb(R)^(n)$. Let $x_(0) \in E$ be a stationary point and $$\displaystyle Q_(x_(0)) \left(h\right) \equiv \sum_(i=1)^n \sum_(j=1) ^n \frac(\partial^(2) f)(\partial x_(i) \partial x_(j)) \left(x_(0)\right)h^(i)h^(j).$$ Then
- if $Q_(x_(0))$ – , then the function $f$ at the point $x_(0)$ has a local extremum, namely, a minimum if the form is positive definite, and a maximum if the form is negative definite;
- if the quadratic form $Q_(x_(0))$ is undefined, then the function $f$ at the point $x_(0)$ has no extremum.
Let's use the expansion according to Taylor's formula (12.7 p. 292). Considering that the first order partial derivatives at the point $x_(0)$ are equal to zero, we obtain $$\displaystyle f \left(x_(0)+h\right)−f \left(x_(0)\right) = \ frac(1)(2) \sum_(i=1)^n \sum_(j=1)^n \frac(\partial^(2) f)(\partial x_(i) \partial x_(j)) \left(x_(0)+\theta h\right)h^(i)h^(j),$$ where $0<\theta<1$. Обозначим $\displaystyle a_{ij}=\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}} \left(x_{0}\right)$. В силу теоремы Шварца (12.6 стр. 289-290) , $a_{ij}=a_{ji}$. Обозначим $$\displaystyle \alpha_{ij} \left(h\right)=\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}} \left(x_{0}+\theta h\right)−\frac{\partial^{2} f}{\partial x_{i} \partial x_{j}} \left(x_{0}\right).$$ По предположению, все непрерывны и поэтому $$\lim_{h \rightarrow 0} \alpha_{ij} \left(h\right)=0. \left(1\right)$$ Получаем $$\displaystyle f \left(x_{0}+h\right)−f \left(x_{0}\right)=\frac{1}{2}\left.$$ Обозначим $$\displaystyle \epsilon \left(h\right)=\frac{1}{|h|^{2}}\sum_{i=1}^n \sum_{j=1}^n \alpha_{ij} \left(h\right)h_{i}h_{j}.$$ Тогда $$|\epsilon \left(h\right)| \leq \sum_{i=1}^n \sum_{j=1}^n |\alpha_{ij} \left(h\right)|$$ и, в силу соотношения $\left(1\right)$, имеем $\epsilon \left(h\right) \rightarrow 0$ при $h \rightarrow 0$. Окончательно получаем $$\displaystyle f \left(x_{0}+h\right)−f \left(x_{0}\right)=\frac{1}{2}\left. \left(2\right)$$ Предположим, что $Q_{x_{0}}$ – положительноопределенная форма. Согласно лемме о положительноопределённой квадратичной форме (12.8.1 стр. 295, Лемма 1) , существует такое положительное число $\lambda$, что $Q_{x_{0}} \left(h\right) \geqslant \lambda|h|^{2}$ при любом $h$. Поэтому $$\displaystyle f \left(x_{0}+h\right)−f \left(x_{0}\right) \geq \frac{1}{2}|h|^{2} \left(λ+\epsilon \left(h\right)\right).$$ Так как $\lambda>0$, and $\epsilon \left(h\right) \rightarrow 0$ for $h \rightarrow 0$, then right part will be positive for any vector $h$ of sufficiently small length.
So, we have come to the conclusion that in a certain neighborhood of the point $x_(0)$ the inequality $f \left(x\right) >f \left(x_(0)\right)$ holds if only $x \neq x_ (0)$ (we put $x=x_(0)+h$\right). This means that at the point $x_(0)$ the function has a strict local minimum, and thus the first part of our theorem is proved.
Let us now assume that $Q_(x_(0))$ is an indefinite form. Then there are vectors $h_(1)$, $h_(2)$ such that $Q_(x_(0)) \left(h_(1)\right)=\lambda_(1)>0$, $Q_ (x_(0)) \left(h_(2)\right)= \lambda_(2)<0$. В соотношении $\left(2\right)$ $h=th_{1}$ $t>$0. Then we get $$f \left(x_(0)+th_(1)\right)−f \left(x_(0)\right) = \frac(1)(2) \left[ t^(2) \ lambda_(1) + t^(2) |h_(1)|^(2) \epsilon \left(th_(1)\right) \right] = \frac(1)(2) t^(2) \ left[ \lambda_(1) + |h_(1)|^(2) \epsilon \left(th_(1)\right) \right].$$ For sufficiently small $t>0$, the right-hand side is positive. This means that in any neighborhood of the point $x_(0)$ the function $f$ takes values $f \left(x\right)$ greater than $f \left(x_(0)\right)$.
Similarly, we find that in any neighborhood of the point $x_(0)$ the function $f$ takes values less than $f \left(x_(0)\right)$. This, together with the previous one, means that at the point $x_(0)$ the function $f$ does not have an extremum.
Let's consider special case of this theorem for a function $f \left(x,y\right)$ of two variables defined in a certain neighborhood of the point $\left(x_(0),y_(0)\right)$ and having continuous partial derivatives of the first in this neighborhood and second orders. Assume that $\left(x_(0),y_(0)\right)$ is a stationary point and denote $$\displaystyle a_(11)= \frac(\partial^(2) f)(\partial x ^(2)) \left(x_(0) ,y_(0)\right), a_(12)=\frac(\partial^(2) f)(\partial x \partial y) \left(x_( 0), y_(0)\right), a_(22)=\frac(\partial^(2) f)(\partial y^(2)) \left(x_(0), y_(0)\right ).$$ Then the previous theorem takes the following form.
Theorem
Let $\Delta=a_(11) \cdot a_(22) − a_(12)^2$. Then:
- if $\Delta>0$, then the function $f$ has a local extremum at the point $\left(x_(0),y_(0)\right)$, namely, a minimum if $a_(11)>0$ , and maximum if $a_(11)<0$;
- if $\Delta<0$, то экстремума в точке $\left(x_{0},y_{0}\right)$ нет. Как и в одномерном случае, при $\Delta=0$ экстремум может быть, а может и не быть.
Examples of problem solving
Algorithm for finding the extremum of a function of many variables:
- Finding stationary points;
- Find the 2nd order differential at all stationary points
- Using the sufficient condition for the extremum of a function of many variables, we consider the 2nd order differential at each stationary point
- Investigate the function for extremum $f \left(x,y\right) = x^(3) + 8 \cdot y^(3) + 18 \cdot x — 30 \cdot y$.
SolutionLet's find the 1st order partial derivatives: $$\displaystyle \frac(\partial f)(\partial x)=3 \cdot x^(2) - 6 \cdot y;$$ $$\displaystyle \frac(\partial f)(\partial y)=24 \cdot y^(2) — 6 \cdot x.$$ Let's compose and solve the system: $$\displaystyle \begin(cases)\frac(\partial f)(\partial x) = 0\\\frac(\partial f)(\partial y)= 0\end(cases) \Rightarrow \begin(cases)3 \cdot x^(2) - 6 \cdot y= 0\\24 \cdot y^(2) — 6 \cdot x = 0\end(cases) \Rightarrow \begin(cases)x^(2) — 2 \cdot y= 0\\4 \cdot y^(2) — x = 0 \end(cases)$$ From the 2nd equation we express $x=4 \cdot y^(2)$ - substitute it into the 1st equation: $$\displaystyle \left(4 \cdot y^(2)\right )^(2)-2 \cdot y=0$$ $$16 \cdot y^(4) — 2 \cdot y = 0$$ $$8 \cdot y^(4) — y = 0$$ $$y \left(8 \cdot y^(3) -1\right)=0$$ As a result, 2 stationary points are obtained:
1) $y=0 \Rightarrow x = 0, M_(1) = \left(0, 0\right)$;
2) $\displaystyle 8 \cdot y^(3) -1=0 \Rightarrow y^(3)=\frac(1)(8) \Rightarrow y = \frac(1)(2) \Rightarrow x=1 , M_(2) = \left(\frac(1)(2), 1\right)$
Let's check whether the sufficient condition for an extremum is satisfied:
$$\displaystyle \frac(\partial^(2) f)(\partial x^(2))=6 \cdot x; \frac(\partial^(2) f)(\partial x \partial y)=-6; \frac(\partial^(2) f)(\partial y^(2))=48 \cdot y$$
1) For the point $M_(1)= \left(0,0\right)$:
$$\displaystyle A_(1)=\frac(\partial^(2) f)(\partial x^(2)) \left(0,0\right)=0; B_(1)=\frac(\partial^(2) f)(\partial x \partial y) \left(0,0\right)=-6; C_(1)=\frac(\partial^(2) f)(\partial y^(2)) \left(0,0\right)=0;$$
$A_(1) \cdot B_(1) — C_(1)^(2) = -36<0$ , значит, в точке $M_{1}$ нет экстремума.
2) For point $M_(2)$:
$$\displaystyle A_(2)=\frac(\partial^(2) f)(\partial x^(2)) \left(1,\frac(1)(2)\right)=6; B_(2)=\frac(\partial^(2) f)(\partial x \partial y) \left(1,\frac(1)(2)\right)=-6; C_(2)=\frac(\partial^(2) f)(\partial y^(2)) \left(1,\frac(1)(2)\right)=24;$$
$A_(2) \cdot B_(2) — C_(2)^(2) = 108>0$, which means that at point $M_(2)$ there is an extremum, and since $A_(2)>0$, then this is the minimum.
Answer: The point $\displaystyle M_(2)\left(1,\frac(1)(2)\right)$ is the minimum point of the function $f$. - Investigate the function for the extremum $f=y^(2) + 2 \cdot x \cdot y - 4 \cdot x - 2 \cdot y - 3$.
SolutionLet's find stationary points: $$\displaystyle \frac(\partial f)(\partial x)=2 \cdot y - 4;$$ $$\displaystyle \frac(\partial f)(\partial y)=2 \cdot y + 2 \cdot x — 2.$$
Let's compose and solve the system: $$\displaystyle \begin(cases)\frac(\partial f)(\partial x)= 0\\\frac(\partial f)(\partial y)= 0\end(cases) \ Rightarrow \begin(cases)2 \cdot y - 4= 0\\2 \cdot y + 2 \cdot x - 2 = 0\end(cases) \Rightarrow \begin(cases) y = 2\\y + x = 1\end(cases) \Rightarrow x = -1$$
$M_(0) \left(-1, 2\right)$ is a stationary point.
Let's check whether the sufficient condition for the extremum is met: $$\displaystyle A=\frac(\partial^(2) f)(\partial x^(2)) \left(-1,2\right)=0; B=\frac(\partial^(2) f)(\partial x \partial y) \left(-1,2\right)=2; C=\frac(\partial^(2) f)(\partial y^(2)) \left(-1,2\right)=2;$$
$A \cdot B — C^(2) = -4<0$ , значит, в точке $M_{0}$ нет экстремума.
Answer: there are no extremes.
Time limit: 0
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Number of points: 1Does the function $f = 4 + \sqrt((x^(2)+y^(2))^(2))$ have an extremum
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Number of points: 1Investigate the function $f$ for extrema: $f=e^(x+y)(x^(2)-2 \cdot y^(2))$
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For a function of one variable y = f(x) at the point x 0 the geometric meaning of the differential means the increment of the ordinate of the tangent drawn to the graph of the function at the point with the abscissa x 0 when moving to a point x 0 + x. And the differential of a function of two variables in this regard is an increment fingerings tangent plane drawn to the surface given by the equation z = f(x, y) , at point M 0 (x 0 , y 0 ) when moving to a point M(x 0 + x, y 0 + y). Let us define a tangent plane to a certain surface:
Df . Plane passing through a point R 0 surfaces S, called tangent plane at a given point, if the angle between this plane and a secant passing through two points R 0 And R(any point on the surface S) , tends to zero when the point R tends along this surface to a point R 0 .
Let the surface S given by the equation z = f(x, y). Then it can be shown that this surface has at the point P 0 (x 0 , y 0 , z 0 ) tangent plane if and only if the function z = f(x, y) is differentiable at this point. In this case, the tangent plane is given by the equation:
z –
z 0
=
+
(6).
§5. Directional derivative, gradient of a function.
Partial derivative functions y=
f(x 1
,
x 2
..
x n )
by variables x 1
,
x 2
. . .
x n express the rate of change of a function in the direction of the coordinate axes. For example, 
is the rate of change of the function according to X 1
– that is, it is assumed that a point belonging to the domain of definition of the function moves only parallel to the axis OH 1
, and all other coordinates remain unchanged. However, it can be assumed that the function can also change in some other direction that does not coincide with the direction of any of the axes.
Consider a function of three variables: u= f(x, y, z).
Let's fix the point M 0 (x 0 , y 0 , z 0 ) and some directed straight line (axis) l, passing through this point. Let M(x, y, z) - an arbitrary point of this line and M 0 M- distance from M 0 before M.
u = f (x, y, z) – f(x 0 , y 0 , z 0 ) – increment of function at a point M 0 .
Let's find the ratio of the increment of the function to the length of the vector 
:
Df . Derivative of a function u = f (x, y, z) towards l at the point M 0 is called the limit of the ratio of the increment of a function to the length of the vector M 0 Mas the latter tends to 0 (or, which is the same thing, as the M To M 0 ):
(1)
This derivative characterizes the rate of change of the function at the point M 0 in the direction l.
Let the axis l
(vector M 0
M)
forms with axes OX,
OY,
OZ angles 
respectively.
Let's denote x-x 0 = 
;
y - y 0 = 
;
z - z 0 = 
.
Then the vector M 0
M = (x
-
x 0
,
y
-
y 0
,
z
-
z 0
)=
and its direction cosines:
;
;
.
(4).
(4) – formula for calculating the directional derivative.
Consider a vector whose coordinates are the partial derivatives of the function u= f(x, y, z) at the point M 0 :
grad u - function gradient u= f(x, y, z) at the point M(x, y, z)
Gradient properties:
Conclusion: function gradient length u=
f(x,
y,
z)
– is the most possible value 
at this point M(x,
y,
z)
, and the direction of the vector grad
u coincides with the direction of the vector leaving the point M, along which the function changes most rapidly. That is, the direction of the gradient of the function
grad
u
- is the direction of the fastest increase in the function.
DIFFERENTIAL CALCULUS OF FUNCTIONS OF SEVERAL VARIABLES.
Basic concepts and definitions.
When considering functions of several variables, we will limit ourselves to a detailed description of the functions of two variables, since all the results obtained will be valid for functions of an arbitrary number of variables.
If each pair of numbers (x, y) independent of each other from a certain set, according to some rule, is associated with one or more values of the variable z, then the variable z is called function of two variables.
If a pair of numbers (x, y) corresponds to one value z, then the function is called unambiguous, and if more than one, then – polysemantic.
Domain of definition function z is the set of pairs (x, y) for which function z exists.
Neighborhood of a point M 0 (x 0, y 0) of radius r is the set of all points (x, y) that satisfy the condition.
The number A is called limit function f(x, y) as the point M(x, y) tends to the point M 0 (x 0, y 0), if for each number e > 0 there is a number r > 0 such that for any point M(x, y), for which the condition is true
the condition is also true 
.
Write down: 
Let the point M 0 (x 0, y 0) belong to the domain of definition of the function f(x, y). Then the function z = f(x, y) is called continuous at point M 0 (x 0, y 0), if
(1)
and the point M(x, y) tends to the point M 0 (x 0, y 0) in an arbitrary manner.
If at any point condition (1) is not satisfied, then this point is called break point functions f(x, y). This may be in the following cases:
1) The function z = f(x, y) is not defined at the point M 0 (x 0, y 0).
2) There is no limit.
3) This limit exists, but it is not equal to f(x 0 , y 0).
Properties of functions of several variables related to their continuity.
Property. If the function f(x, y, ...) is defined and continuous in a closed and bounded domain D, then in this domain there are at least one point
N(x 0 , y 0 , …), such that for the remaining points the inequality is true
f(x 0 , y 0 , …) ³ f(x, y, …)
as well as point N 1 (x 01, y 01, ...), such that for all other points the inequality is true
f(x 01 , y 01 , …) £ f(x, y, …)
then f(x 0 , y 0 , …) = M – highest value functions, and f(x 01 , y 01 , ...) = m – smallest value functions f(x, y, …) in domain D.
A continuous function in a closed and bounded domain D reaches at least once highest value and once the smallest.
Property. If the function f(x, y, …) is defined and continuous in a closed bounded domain D, and M and m are, respectively, the largest and smallest values of the function in this domain, then for any point m О there is a point
N 0 (x 0 , y 0 , …) such that f(x 0 , y 0 , …) = m.
Simply put, continuous function accepts everything in area D intermediate values between M and m. A consequence of this property can be the conclusion that if the numbers M and m are of different signs, then in the domain D the function vanishes at least once.
Property. Function f(x, y, …), continuous in a closed bounded domain D, limited in this region, if there is a number K such that for all points in the region the inequality is true 
.
Property. If a function f(x, y, …) is defined and continuous in a closed bounded domain D, then it uniformly continuous in this area, i.e. for anyone positive number e there is a number D > 0 such that for any two points (x 1, y 1) and (x 2, y 2) of the region located at a distance less than D, the inequality holds
2. Partial derivatives. Partial derivatives of higher orders.
Let a function z = f(x, y) be given in some domain. Let's take an arbitrary point M(x, y) and set the increment Dx to the variable x. Then the quantity D x z = f(x + Dx, y) – f(x, y) is called partial increment of the function in x.
You can write down
.
Then it's called partial derivative functions z = f(x, y) in x.
Designation: 
The partial derivative of a function with respect to y is determined similarly.
Geometric sense the partial derivative (let's say) is the tangent of the angle of inclination of the tangent drawn at the point N 0 (x 0, y 0, z 0) to the section of the surface by the plane y = y 0.
If a function f(x, y) is defined in some domain D, then its partial derivatives will also be defined in the same domain or part of it.
We will call these derivatives first order partial derivatives.
The derivatives of these functions will be second order partial derivatives.
Continuing to differentiate the resulting equalities, we obtain partial derivatives of higher orders.
Partial derivatives of the form 
etc. are called mixed derivatives.
Theorem. If the function f(x, y) and its partial derivatives are defined and continuous at the point M(x, y) and its vicinity, then the following relation is true:
Those. partial derivatives of higher orders do not depend on the order of differentiation.
Higher order differentials are defined similarly.
…………………
Here n is the symbolic power of the derivative, which is replaced by the real power after raising the expression in brackets to it.
Full differential. Geometric meaning full differential. Tangent plane and normal to the surface.
The expression is called full increment functions f(x, y) at some point (x, y), where a 1 and a 2 are infinitesimal functions for Dх ® 0 and Dу ® 0, respectively.
Full differential function z = f(x, y) is called the main linear part with respect to Dx and Dу of the increment of the function Dz at the point (x, y).
For a function of an arbitrary number of variables:
Example 3.1. Find the complete differential of the function.
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