Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
Warm-up: Imagine you are standing on the side of a tall hill. List three things you would want to know about your position.
On your Mark: Place your surface on the grid. Label the \(x\) and \(y\) directions on the grid and surface. Measure the slope in the direction of greatest increase of the surface at the blue dot. Include units.
Get Set: The surface's height \(h\) is a function of \(x\) and \(y\), giving \(h = h(x,y)\). At the blue dot, measure both \(\frac{\partial h}{\partial x}\) and \(\frac{\partial h}{\partial y}\). Then form the vector \(\frac{\partial h}{\partial x}\,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y}\,\boldsymbol{\hat{y}}\). Include units.
Go: At the blue dot, which way does your vector \(\frac{\partial h}{\partial x}\,\boldsymbol{\hat{x}} + \frac{\partial h}{\partial y}\,\boldsymbol{\hat{y}}\) point on the surface?
Challenge: Rotate the surface \(30^\circ\) on the grid and redraw the \(x\) and \(y\) directions on your surface. Which of your answers to On your Mark, Get Set, and Go remain the same?