Level curves and the gradient vector
The following plot shows the surface defined by (in blue). The circle of different colours in the -plane are the level curves of the function (the points where ) for . The points on the surface above those level curves are indicated in the same colour. The red arrow in the -plane is a unit vector pointing in the direction of the gradient vector at the point . The black arrows are the tangent vectors of the level curve , which goes through the point .
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 going through the point .
- 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 is equal, the level curves are not equidistant to each other, the steeper the surface, the closer they are.