Preliminary Example. Let $T = f(x,y) = \frac{1}{2}(x^2 + y^2)$ be
the temperature, in degrees Celsius, at a point $(x, y)$ on a metal
plate, where $x$ and $y$ are measured in cm. A contour plot for
the function $f$ is shown to the right.
(a) Find the minimum temperature of the plate along the
line $x + y = 4$ and the minimum temperature of the plate
along the line $x + y = -2$.
(b) Let $g(x, y) = x + y$, and note that the lines $x + y = 4$ and $x + y = -2$ from part (a) are level curves of $g$. Draw
the gradient vectors $\nabla f$ and $\nabla g$ at the two points from part (a) where the minimum temperature is achieved.
(c) Examine the point on each of the two lines where the minimum temperature is achieved. How are the level
curves of $f$ and $g$ related? How are the vectors $\nabla f$ and $\nabla g$ related?
