## Multivariate bowl: Demo

In this interactive demo, we have plotted the multivariate bowl in three panels.

The larger panel shows the contour plot of the function.
This means, the color band is indicative of the value of \( f(x,y) \).
The variable \( x \) is plotted on the horizontal X-axis.
The variable \( y \) is plotted on the vertical Y-axis.

The remaining two panels show a *slice* of the function if we fix the value along either axis.
This provides looking into the function from the side.
For example, \( f(x,y) \) with \( y \) held constant means the spectator is analyzing the function standing below the chart looking at a particular slice along the Y-axis.

The slice locations along both axes can be changed by moving the interactive orange circle on the contour plot. The slices can be imagined as vertical cuts made into the function along the two dotted lines; one along each axis.
The location of the orange circle is also highlighted on the slice plots as a red dot.

For this particular chart, note that the function achieves a minimum at \( x = 0, y = 0 \).
The function increases in value along all other directions.
By looking at the bands on the contour plot, note that the increase in the function is slower near the center, and gets quite steep as you move away from \( x = 0, y = 0 \).
This can also be verified from the slice plots.

Finally, note that no matter where you move slices, the nature of the function remains the same, a univariate cup.