Plot of f(x,y) = 8xy

Function from page 93 slides

generates two vectors of values for x and y where \(0<x<y<1\)

x <- seq(0.000001, .999998, length= 30)
y <- seq(0.000002, .999999, length= 30)
#gives TRUE if any element into those vectors satisfies the equation
any(x>y)

## [1] FALSE

func <- function (x, y) 8*x*y
domain <- outer(x, y, '<')
z <- outer(x, y, func)
Fxy <- ifelse(domain == F, 0, z) #set 0 where x < y

wireframe(Fxy, drape=T, 
          xlab = 'x',
          ylab = 'y',
          zlab = 'fx',
          main = 'f(x,y)=c(x,y)',
          col.regions=rainbow(100))

\(0\leqslant x \leqslant y \leqslant 1\)

func <- function (x, y) 2*x^2*y^2 - x^4
z <- outer(x, y, func)
Fxy <- ifelse(domain == F, 0, z)
wireframe(Fxy, drape=T, 
          xlab = 'x',
          ylab = 'y',
          zlab = 'fx',
          main = 'F(x,y)= 2 (xy)^2 - x^4',
          col.regions=rainbow(100))

if \(0\leqslant y \leqslant 1 \lor x>y\) then \(F(x,y) = y^4\)

if \(0\leqslant x \leqslant 1 \lor y>1\)

func <- function (x, y) 2*x^2 - x^4
z <- outer(x, y, func)
Fxy <- ifelse(domain == F, 0, z)
wireframe(Fxy, drape=T, 
          xlab = 'x',
          ylab = 'y',
          zlab = 'fx',
          main = 'F(x,y)= 2 x^2 - x^4',
          col.regions=rainbow(100))