Sketching the Gradient of a Scalar Field: How to Implement and Interpret?

[CSNabeel]

Homework Statement

Calculate the gradient of the scalar field f(x,y) = x^{2} - y^{2} . Sketch the gradient for a few point on two straight lines y = x and y = -x on the plane and comment on the properties of the sketch.

Homework Equations

The Attempt at a Solution

So I worked out the gradient to be:

f = 2xi - 2yj

and then I did this for the point

x y x -y
-2 -2 -2 2
-1 -1 -1 1
0 0 0 0
1 1 1 -1
2 2 2 -2

but then I got confused on how to implement the gradient to this to do the sketch! Help would be much appreciated

 

[re444]

The sketch would be something like this:

In each point on the plane, (x,y), there is a gradient vector as you said, 2xi - 2yj . these vectors point to the direction in the function's domain, which the main function has the greatest increase in its value.

 

[HallsofIvy]

At each (x,y) draw a vector having x-component 2x and y-component -2y. That is, go to the right 2x and down 2y (assuming x and y are positive, of course).