CoCalc Shared FilesLevel curves and the gradient vector.sagewsOpen in CoCalc with one click!
Authors: Jeroen Wouters, yang shi
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Level curves and the gradient vector

The following plot shows the surface defined by z=f(x,y)=x2+y2z = f(x,y) = x^2 + y^2 (in blue). The circle of different colours in the (x,y)(x,y)-plane are the level curves of the function (the points where f(x,y)=Cf(x,y) = C) for C=1,2,3,4,5C = 1,2,3,4,5. The points on the surface ff above those level curves are indicated in the same colour. The red arrow in the (x,y)(x,y)-plane is a unit vector pointing in the direction of the gradient vector f(1,1)\nabla f(1,1) at the point (1,1)(1,1). The black arrows are the tangent vectors of the level curve C=2C=2, which goes through the point (1,1)(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=2C=2 going through the point (1,1)(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 CC is equal, the level curves are not equidistant to each other, the steeper the surface, the closer they are.
3D rendering not yet implemented