-functions of two variables
-graph of a function
-level curves with calue „c”
-level curves with respect to „x” or „y”
-partial differentation
-partial derivative of f with respect to „x”
- ----------„ -------------- with respect to „y”
-Theorem of continous partial derivatives
A function of two variables is a rule which assigns a number f(x,y) to each point (x,y) of a
domain int he xy plain.
Example: f(x,y) = x3-3xy
The graph of a function f(x,y) of two variables consist of all point (x,y,z) in space such that
(x,y) is int he domain of the function and z=f(x,y)
Example: z=x3-3xy2
Let be a function of two variables and let „c” be a constant. The set of all (x,y) int he plane
such that f(x,y)=c is called a level curve of f with calue „c”.
Example: x3-3xy2=-0.8
Let f be a funtion of two variables and (a,b)EDf. The level curve with respect to x is the
graph of f1(x)=f(x,b), (x,b)ED f function of one variable.
The level curve with respect to y is the graph of f2(y)=f(a,y) (a,y)ED f function of one
variable.
These are the traces of the surface.
If f is a funtion of two variables, to calculate the partial derivative with respect to a certain
variable, treat the remaining variable as a constant and differentiate as usual by using the rules
of variable calculus.
Remember for the differentation of a constant:
(1) c’=0
(2) (cf)’=c*f’
If z=f(x,y) is a function of two variables, the partial derivative of f with respect to x is
denoted
f’x=deltaz/deltax = limchangingx->0 f(x+changingx,y) – f(x,y) / changingx
If z=f(x,y) is a funtion of two variables, the partial derivative of f with respect to y is
denoted.
f’y=deltaz/deltay=limchanging y->0 f (x,y +changing y) – f(x,y) /changing y
..túl hosszú a következő ez vmi second order partial derivatives
Theorem: If z=f(x,y) has continous partial derivatives, then the mixed partial derivatives are
equal: f”xy=f”yx