Level curves and the gradient vector.sagews

Level curves and the gradient vector

The following plot shows the surface defined by $z = f(x,y) = x^2 + y^2$ (in blue). The circle of different colours in the $(x,y)$-plane are the level curves of the function (the points where $f(x,y) = C$) for $C = 1,2,3,4,5$. The points on the surface $f$ above those level curves are indicated in the same colour. The red arrow in the $(x,y)$-plane is a unit vector pointing in the direction of the gradient vector $\nabla f(1,1)$ at the point $(1,1)$. The black arrows are the tangent vectors of the level curve $C=2$, which goes through the point $(1,1)$.

You can rotate and zoom the plot by dragging and scrolling with your mouse.

Note the following about this graph:

• The gradient vector is orthogonal to the level curve $C=2$ going through the point $(1,1)$.

• The gradient vector points in the direction of the steepest slope of the blue surface above that point.

• The gradient and tangent vectors are orthogonal to each other.

• Even though the difference between the subsequent constants $C$ is equal, the level curves are not equidistant to each other, the steeper the surface, the closer they are.

3D rendering not yet implemented