Building the level and gradient curves of a function f(x,y).Recall the theorems from calculus 3: Theorem 1: If $z=f(x,y)$ is differentiable with nonzero gradient, then the gradient points in the direction of greatest increase of $f$.Theorem 2: If $g(t)=(x(t),y(t))$ is a smooth path, $z=f(x,y)$ is differentiable with nonzero gradient and the velocity $g'(t)$ is orthogonal with the gradient $\nabla f $ at $g(t)$, then $f(g(t))$ is constant, that is, the range of $g$ lies in a level curve of $f$. We can use these theorems to build approximations to the level and gradient curves of $f$ through the point $(x_0,y_0)$. It's important that the gradient of $f$ be nonzero in a neighborhood of $(x_0,y_0)$. For then we can build the ascent (descent) curve of $f$ starting from $(x_0,y_0)$, by choosing a small positive step size $dt=1/N$, and proceed by stepping $(x_{n+1},y_{n+1})=(x_n,y_n)+(-)\dfrac{\nabla f(x_n,y_n)}{|\nabla f(x_n,y_n)|}\,dt$ for $n=0,1,2, \cdots, N$. Similarly, for the level curve (just rotating the gradient vector 90 degrees and then stepping). This method (which is just the basic Euler method for numerical solution of a differential equation) works pretty well to obtain a local image of the level and gradient curves of $f$. By choosing a list of starting points $(x_0,y_0)$, we can piece together these patches to get a bigger picture. Try the Sagecell interact below to experiment with this. If you choose the 3d option t3d, give the image time to form before manipulating it. A Sagecell interact to generate a level-gradient curve patch.fun: an expression in the variables x and y pts: the starting points (x0,y0) of the grid grid: controls the number of gridlines in the display. Smaller number is more gridlines dist: controls the size of the grid dt: controls the step size clrs: 0 = black, 1 = red, 2 = green t3d: check to get 3d image of graph of f |