13b94d…: “Thanks to the rainy book”

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Let $F(x,y)=(x-y)^2$ . Determine the equations and shapes of the cross-sections when $x=0$ , $y=0$ , $x=y$ , and describe the level curves. $\text {When $x=0$, the cross-sections are given by } z=\answer {y^2}$ $\text {When $y=0$, the cross-sections are given by } z=\answer {x^2}$ $\text {When $x=y$, the cross-sections are given by } z=\answer {0}$ The level curves are best described as:

All the level curves will satisfy the equation $(x-y)^2=c$ , for some $c\ge 0$ .

Therefore, a level curve corresponding to an output $c$ is given by

$x-y= \sqrt {c}$ or $x-y= -\sqrt {c}$ , which is equivalent to

$y=x- \sqrt {c}$ or $y=x+\sqrt {c}$ .

Four level curves, corresponding to output values $c=0$ , $c=1$ , $c=4$ , and $c=9$ , are shown in the picture below.

diamonds squares circles lines of slope $1$ hyperbolas