Functions of two variables Graph of a function

plot of graph of f(x, y) = −(cos(x) + cos(y)), showing gradient projected on bottom plane.

the graph of trigonometric function

f
(
x
,
y
)
=
sin
⁡
(

x

2


)
cos
⁡
(

y

2


)



{\displaystyle f(x,y)=\sin(x^{2})\cos(y^{2})\,}

is

{
(
x
,
y
,
sin
⁡
(

x

2


)
cos
⁡
(

y

2


)
)
:
x

 and 

y

 are real numbers

}
.


{\displaystyle \{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ , }}y{\text{ real numbers}}\}.}

if set plotted on 3 dimensional cartesian coordinate system, result surface (see figure).

oftentimes helpful show graph, gradient of function , several level curves. level curves can mapped on function surface or can projected on bottom plane. second figure shows such drawing of graph of function:

f
(
x
,
y
)
=
−
(
cos
⁡
(

x

2


)
+
cos
⁡
(

y

2


)

)

2





{\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}\,}