12.1

  1. Describe in words and sketch the level curves for the function and given c values.

\[f\left( x,y \right) ={ x }^{ 2 }-{ y }^{ 2 };c=-1,0,1\] We start by utilizing the C=0 value. \[c=0...->...f\left( x,y \right) ={ x }^{ 2 }-{ y }^{ 2 }=c=0\] \[y=\pm \sqrt { { x }^{ 2 } } =x\] \[c=-1...->...f\left( x,y \right) ={ x }^{ 2 }-{ y }^{ 2 }=c=-1\] \[y=\pm \sqrt { { x }^{ 2 }-1 } \] \[c=1...->...f\left( x,y \right) ={ x }^{ 2 }-{ y }^{ 2 }=c=1\] \[y=\pm \sqrt { { x }^{ 2 }+1 }\]

x<-seq(-10,20,0.02)
y<-seq(-10,10,0.02)
xy <- data.frame(expand.grid(x=x, y=y))

q16 <- xy$x^2-xy$y^2

z<- data.frame(xy,q16)

ggplot(z, aes(x,y,z=q16))+
  geom_raster(aes(fill=q16))+
  geom_contour(breaks = c(-1,0,1))