Functions of two variables: \(z\left( {x,y} \right),\) \(f\left( {x,y} \right),\) \(g\left( {x,y} \right),\) \(h\left( {x,y} \right),\) \(F\left( {x,y} \right)\)

Arguments (independent variables): \(x,\) \(y,\) \(t\)

Arguments (independent variables): \(x,\) \(y,\) \(t\)

Small changes in \(x\), \(y\), \(z\), respectively: \(\Delta x,\) \(\Delta y,\) \(\Delta z\)

Determinant: \(D\)

Determinant: \(D\)

- The concept of a function of one variable can be easily generalized to the case of two or more variables. In the case of a function of two variables, we consider the set of ordered pairs \(\left( {x,y} \right)\) where the values of \(x\) and \(y\) belong to the sets \(x \in X\), \(y \in Y\). If there is a law according to which each pair \(\left( {x,y} \right)\) is assigned to a single value of \(z\), then we say about a function of two variables. Usually, this function is denoted by

\(z = z\left( {x,y} \right),\) \(z = f\left( {x,y} \right),\) \(z = F\left( {x,y} \right),\) etc.

Similarly, we define a function of \(n\) variables. - First order partial derivatives

For functions of several variables, we introduce the notion of partial derivative of the first order, that is the derivative of one of the variables provided that all other variables are held constant. For example, for a two-variable function \(z = f\left( {x,y} \right)\), we can consider the partial derivatives with respect to the variable \(x\) and \(y\).

They are designated as follows:

\({\large\frac{{\partial z}}{{\partial x}}\normalsize} = {\large\frac{{\partial f}}{{\partial x}}\normalsize} =\) \( {z’_x} = {f’_x},\) \({\large\frac{{\partial z}}{{\partial y}}\normalsize} =\) \({\large\frac{{\partial f}}{{\partial y}}\normalsize} =\) \({z’_y} = {f’_y}\) - Second order partial derivatives

\({\large\frac{\partial }{{\partial x}}\normalsize}\left( {{\large\frac{{\partial f}}{{\partial x}}}\normalsize} \right) = {\large\frac{{{\partial ^2}f}}{{\partial {x^2}}}\normalsize} = {f^{\prime\prime}_{xx}},\;\)

\({\large\frac{\partial }{{\partial y}}\normalsize}\left( {{\large\frac{{\partial f}}{{\partial y}}}\normalsize} \right) = {\large\frac{{{\partial ^2}f}}{{\partial {y^2}}}\normalsize} = {f^{\prime\prime}_{yy}},\)

\({\large\frac{\partial }{{\partial y}}\normalsize}\left( {{\large\frac{{\partial f}}{{\partial x}}}\normalsize} \right) = {\large\frac{{{\partial ^2}f}}{{\partial y\,\partial x}}\normalsize} = {f^{\prime\prime}_{xy}},\;\)

\({\large\frac{\partial }{{\partial x}}\normalsize}\left( {{\large\frac{{\partial f}}{{\partial y}}}\normalsize} \right) = {\large\frac{{{\partial ^2}f}}{{\partial x\,\partial y}}\normalsize} = {f^{\prime\prime}_{yx}}.\)

If the mixed partial derivatives are continuous, they do not depend on the order in which the derivatives are taken:

\({\large\frac{{{\partial ^2}f}}{{\partial y\,\partial x}}\normalsize} = {\large\frac{{{\partial ^2}f}}{{\partial x \,\partial y}}\normalsize}\) - Differentiation of a composite function of two variables

If \(f\left( {x,y} \right) = g\left( {h\left( {x,y} \right)} \right),\) where \(g\) is a function of one variable \(h\), then the partial derivatives are given by

\({\large\frac{{\partial f}}{{\partial x}}\normalsize} = g’\left( {h\left( {x,y} \right)} \right){\large\frac{{\partial h}}{{\partial x}}\normalsize},\;\) \({\large\frac{{\partial f}}{{\partial y}}\normalsize} = g’\left( {h\left( {x,y} \right)} \right){\large\frac{{\partial h}}{{\partial y}}\normalsize}.\)

If \(h\left( t \right) = f\left( {x\left( t \right),y\left( t \right)} \right)\), the derivative is expressed by the formula

\(h’\left( t \right) =\) \({\large\frac{{\partial f}}{{\partial x}}\frac{{dx}}{{dt}}\normalsize} + {\large\frac{{\partial f}}{{\partial y}}\frac{{dy}}{{dt}}\normalsize.}\)

If \(z = f\left( {x\left( {u,v} \right),y\left( {u,v} \right)} \right)\), then the partial derivatives are determined by the expressions

\({\large\frac{{\partial z}}{{\partial u}}\normalsize} = {\large\frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial u}}\normalsize} + {\large\frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial u}}\normalsize},\;\)

\({\large\frac{{\partial z}}{{\partial v}}\normalsize} = {\large\frac{{\partial f}}{{\partial x}}\frac{{\partial x}}{{\partial v}}\normalsize} + {\large\frac{{\partial f}}{{\partial y}}\frac{{\partial y}}{{\partial v}}\normalsize}.\) - Small changes of a function

\(\Delta z \approx {\large\frac{{\partial f}}{{\partial x}}\normalsize}\Delta x \;+\) \({\large\frac{{\partial f}}{{\partial y}}\normalsize}\Delta y\) - Local maxima and minima

A function \(f\left( {x,y} \right)\) has a local maximum at a point \(\left( {{x_0},{y_0}} \right)\) if \(f\left( {x,y} \right) \lt f\left( {{x_0},{y_0}} \right)\) for all \(\left( {x,y} \right)\) sufficiently close to \(\left( {{x_0},{y_0}} \right)\).

Similarly, a function \(f\left( {x,y} \right)\) has a local minimum at a point \(\left( {{x_0},{y_0}} \right)\) if \(f\left( {x,y} \right) \gt f\left( {{x_0},{y_0}} \right)\) for all \(\left( {x,y} \right)\) sufficiently close to \(\left( {{x_0},{y_0}} \right)\).

The points of maximum and minimum of a function are called the extreme points. - Stationary and critical points

The points at which all partial derivatives are zero are called stationary points. For a function of two variables, the stationary points can be found from the system of equations

\(\large\frac{{\partial f}}{{\partial x}}\normalsize = \large\frac{{\partial f}}{{\partial y}}\normalsize = 0.\)

The points of local maximum and minimum are stationary points. The stationary points along with the points of the domain in which the partial derivatives do not exist form the set of critical points. - Saddle point

A stationary point which is neither a local maximum nor a local minimum is called a saddle point. - Second derivative test for stationary points

Let \(\left( {{x_0},{y_0}} \right)\) be a stationary point (i.e. the partial derivatives are equal to zero at this point). Consider the determinant composed of the values of the second order partial derivatives at this point:

\(D \text{ = }\) \(\left| {\begin{array}{*{20}{c}} {\large\frac{{{\partial ^2}f}}{{\partial {x^2}}}\normalsize\left( {{x_0},{y_0}} \right)} & {\large\frac{{{\partial ^2}f}}{{\partial y\,\partial x}}\normalsize\left( {{x_0},{y_0}} \right)}\\ {\large\frac{{{\partial ^2}f}}{{\partial x\,\partial y}}\normalsize\left( {{x_0},{y_0}} \right)} & {\large\frac{{{\partial ^2}f}}{{\partial {y^2}}}\normalsize\left( {{x_0},{y_0}} \right)} \end{array}} \right|.\)

If \(D \gt 0\) and the partial derivative \({\large\frac{{{\partial ^2}f}}{{\partial {x^2}}}\normalsize}\left( {{x_0},{y_0}} \right) \gt 0,\) then \(\left( {{x_0},{y_0}} \right)\) is a point of local minimum.

If \(D \gt 0\) and the partial derivative \({\large\frac{{{\partial ^2}f}}{{\partial {x^2}}}\normalsize}\left( {{x_0},{y_0}} \right) \lt 0,\) then \(\left( {{x_0},{y_0}} \right)\) is a point of local maximum.

If \(D \lt 0\), then \(\left( {{x_0},{y_0}} \right)\) is a saddle point.

If \(D = 0\), then this test fails. - Tangent plane

The equation of the tangent plane to the surface \(z = f\left( {x,y} \right)\) at the point \(\left( {{x_0},{y_0},{z_0}} \right)\) is given by

\(z – {z_0} =\) \({\large\frac{{\partial f}}{{\partial x}}\normalsize}\left( {{x_0},{y_0}} \right)\left( {x – {x_0}} \right) \;+\) \({\large\frac{{\partial f}}{{\partial y}}\normalsize}\left( {{x_0},{y_0}} \right)\left( {y – {y_0}} \right).\) - Normal to surface

The equation of the normal to the surface \(z = f\left( {x,y} \right)\) at the point \(\left( {{x_0},{y_0},{z_0}} \right)\) is written as

\({\large\frac{{x – {x_0}}}{{\frac{{\partial f}}{{\partial x}}\left( {{x_0},{y_0}} \right)}}\normalsize} = {\large\frac{{y – {y_0}}}{{\frac{{\partial f}}{{\partial y}}\left( {{x_0},{y_0}} \right)}}\normalsize} =\) \({\large\frac{{z – {z_0}}}{{ – 1}}\normalsize}.\)